a comparison of statistical disclosure control methods: multiple

Report Issued: January 23, 2013 Disclaimer: This report is released to inform interested parties of research and to encourage discussion. The views expressed are those of the authors and not necessarily those of the U.S. Census Bureau.

RESEARCH REPORT SERIES (Statistics #2013-02)

A Comparison of Statistical Disclosure Control Methods: Multiple Imputation Versus

Noise Multiplication

Martin Klein Thomas Mathew

Bimal Sinha

Center for Statistical Research & Methodology Research and Methodology Directorate

U.S. Census Bureau Washington, D.C. 20233

A Comparison of Statistical Disclosure Control Methods:Multiple Imputation Versus Noise Multiplication

Martin KLEIN, Thomas MATHEW and Bimal SINHA

When statistical agencies release microdata to the public, a major concern is the control

of disclosure risk, while ensuring utility in the released data. Often some statistical disclosure

control methods such as data swapping, multiple imputation, top coding, and perturbation

with random noise, are applied before releasing the data. This article provides a compre-

hensive comparison of two methods, namely, multiple imputation and noise multiplication,

for drawing inference about some useful parameters under the exponential, normal and log-

normal models. The comparison is provided under two scenarios: (1) the entire data set is

replaced by multiply imputed or noise multiplied data, and (2) only the top part of the data

is similarly replaced. The latter scenario arises, for example, when top coding is used for

disclosure control, especially with income data. Methodology is developed for the analysis of

noise multiplied data under both scenarios. Under the situation where only the large values

in the dataset are noise multiplied, data analysis methods are developed and compared under

two types of data releases: (i) each released value includes an indicator of whether or not

it has been noise perturbed, and (ii) no such indicator is provided. The comparison study

shows that data analyses under the multiple imputation and noise multiplication methods

can provide similar results in terms of accuracy of statistical inferences; and that noise multi-

Martin Klein (E-mail: [email protected]) and Thomas Mathew (E-mail:[email protected]) are Research Mathematical Statisticians in the Center for Statistical Researchand Methodology, U.S. Census Bureau, Washington, DC 20233. Bimal Sinha (E-mail: [email protected]) isResearch Mathematical Statistician in the Center for Disclosure Avoidance Research, U.S. Census Bureau,Washington, DC 20233. Mathew and Sinha are also Professors in the Department of Mathematics andStatistics, University of Maryland Baltimore County, Baltimore, MD 21250. The authors are grateful toJerry Reiter for providing the data used in Section 7. They are thankful to Joseph Schafer, Eric Slud,Yves Thibaudeau, Tommy Wright and Laura Zayatz for encouragement, and to Paul Massell for somehelpful comments. This article is released to inform interested parties of ongoing research and to encouragediscussion of work in progress. The views expressed are those of the authors and not necessarily those ofthe U.S. Census Bureau.

1

plication can provide either more or less accuracy than multiple imputation by appropriately

adjusting the variance of the noise generating distribution. Extensive simulation results pro-

vide guidance as to how the noise variance affects accuracy of inference in several parametric

settings. A comparison using data from the 2000 U.S. Current Population Survey highlights

the similarities of the methods. Detailed tables summarizing simulation results and some

technical derivations are available online as supplementary material.

KEY WORDS: Customized noise distribution; EM algorithm; Partially synthetic data; Syn-

thetic data; Tuning parameter.

1. INTRODUCTION

When survey organizations and statistical agencies such as the U.S. Census Bureau re-

lease microdata to the public, a major concern is the control of disclosure risk, while si-

multaneously ensuring quality and utility of the released data. Very often some popular

statistical disclosure control methods such as data swapping, multiple imputation (MI), top

coding/bottom coding (especially for income data), and multiplication with random noise,

are applied before releasing the data. Multiple imputation has been in existence for some

time as a viable methodology to handle missing data (see Rubin, 1987); following the initial

proposal by Rubin (1993), in a series of papers (e.g., Drechsler and Reiter, 2010; Raghu-

nathan, Reiter, and Rubin, 2003; Reiter, 2003, 2004, 2005a, 2005b) Reiter and his colleagues

expanded its scope and provided a solid and rigorous foundation for its use so much so that

statistical agencies can now employ this method for sensitive data protection while data

users can carry out the required inference in a valid way. When MI is applied for statis-

tical disclosure control, the multiply imputed data that are ultimately released are usually

referred to as synthetic data. More recently, multiple imputation has been cleverly used by

2

An and Little (2007) as an alternative to top coding. Recall that top coding consists of

censoring the top part of the data above a specified threshold, and is commonly used in the

context of income data so that the identity of those in the top income bracket is protected.

We refer to the recent monograph by Drechsler (2011) for a detailed discussion of multiple

imputation as a tool for disclosure control. Noise perturbation by addition or multiplication

has also been advocated by some statisticians as a possible data confidentiality protection

mechanism (Hwang, 1986; Little, 1993; Kim and Winkler, 2003); recently there has a been

a renewed interest on this topic (Nayak, Sinha and Zayatz, 2011; Sinha, Nayak and Zayatz,

2011).

In this paper we provide a comprehensive comparison of the above two methods, namely,

multiple imputation (MI) and noise multiplication, for drawing inference about some use-

ful parameters under three parametric models. Both of these methods have been applied

to Census Bureau data products. For instance, multiply imputed synthetic data prod-

ucts have been developed and are available for the following: OnTheMap database of

commuting patterns (Machanavajjhala et al., 2008), the Longitudinal Business Database

(Kinney et al., 2011), the Survey of Income and Program Participation (Abowd et al.,

2006), and American Community Survey group quarters data (Rodriguez, 2007; Hawala,

2008). Regarding application of noise multiplied data, the first public use microdata sample

(PUMS) produced from the Survey of Business Owners (SBO) was released in August 2012

(http://www.census.gov/econ/sbo/), and noise multiplication was employed for confiden-

tiality protection of some variables. Here each record corresponds to a business surveyed in

the 2007 SBO, and a number of variables are provided relating to firm size, business char-

acteristics, and business owner characteristics. In this data product, a number of steps are

taken to protect confidentiality of businesses, and the variables relating to receipts, payroll,

and employment are rounded and multiplied by random noise prior to release (2007 Survey

of Business Owners (SBO) Public Use Microdata Sample (PUMS) Data Users Guide, 2012).

3

Given that both multiple imputation and noise multiplication are actually being employed

for disclosure limitation, a comparison between these two is clearly of practical interest. It

should however be noted that while methodology for data analysis is now well developed

in the context of multiple imputation, such is not the case for noise multiplication. Thus a

major goal of this work is the development of rigorous data analysis methodology for noise

multiplied data. We have considered three popular univariate parametric models: exponen-

tial, normal, and lognormal, and have dealt with the problem of estimation of some standard

parametric functions such as mean, variance and percentiles. We show that data analyses

under the multiple imputation method as well as under the noise multiplication method can

provide similar results in terms of accuracy of inferences, and under noise multiplication the

inferences can be made either more or less accurate through adjustment of the variance of

the noise generating distribution; in this sense, noise multiplication does offer some flexibility

compared to multiple imputation.

When multiple imputation (MI) and noise multiplication (NM) are applied to the entire

data at hand, we will refer to the scenario as total MI and fully NM case. Often times there

are situations when a part of the data is sensitive and must not be released while the rest

of the data can be used/released without any compromise. This is the set up of top coding

mentioned above, where values above a certain threshold C are suppressed and only the

number of values in the data set above C are reported along with the actual values below C.

This is precisely the scenario considered by An and Little (2007), and they have developed

data analysis methods based on multiple imputation of the data above C, in combination

with the original values below C. We can also consider noise multiplication of the values

above C, and release the data consisting of such noise multiplied values, along with the

actual observations below C. Again, we will provide a comparison of An and Little’s (2007)

multiple imputation along with the noise multiplication idea just mentioned, for the three

parametric models mentioned earlier. Under the scenario of noise multiplication of only

4

values above the threshold C, we consider two types of data releases referred to as cases (i)

and (ii). In case (i) data release, each released variable includes an indicator of whether or

not it has been perturbed, while in case (ii), no such indicator is provided. Data analysis

methods for both of these cases are developed in Section 3. Naturally, case (i) data appear to

carry more information than case (ii) data, and so one would expect that case (i) would lead

to more accurate inference than case (ii), perhaps at an increased disclosure risk. The case

(i) data always yield more (or equally) accurate inferences than a top coded sample, whereas

depending on the magnitude of the noise variance, case (ii) data can yield either more or less

accurate inference than a top coded sample; we elaborate and justify these points in Section

6.

Here is the organization of the paper. In Section 2 we provide details of the statistical

analysis for the case of fully noise perturbed data. Section 3 deals with developing statistical

analysis based on noise perturbation of only top-coded data, keeping the rest of the data

(below C) as it is. Appendix A provides proof of a technical result used in Section 3 and

states some related results. Following the work of Reiter (2003) and An and Little (2007),

we provide a brief review of the multiple imputation method for both the total MI and top-

coded data situations in Section 4. Section 5 briefly mentions statistical data analysis under

usual top coding so that we have censored data. Conclusions based on extensive numerical

results are given in Section 6. In Section 7, an application using data from the 2000 U.S.

Current Population Survey is given to illustrate the similarities of the methods. The data

comprise of seventeen (17) variables measured on 51,016 heads of households (see Table 1 in

Drechsler and Reiter, 2010, for a description of ten variables) including age, race, sex and

marital status as key identifiers, and a mix of other categorical and numerical variables with

non-Gaussian distributions, and with large percentages of values equal to zero. We have

selected two such numerical variables with extreme sample sizes, namely, total household

income with sample size = 50,661 and household alimony payment with sample size 206, and

5

have carried out the analysis (after deleting the zero values). It turns out that a lognormal

distribution fits both the datasets well, and the inferences drawn upon the original mean

and variance and the log-scale mean and variance are comparable under the two methods:

multiple imputation and noise multiplication with noise variance not exceeding 10%. In fact,

the inferences closely reproduce those based on the original data. We conclude the paper

with a summary and recommendations in Section 8.

We end this section with two general observations. First, while standard and often op-

timum parametric inference can be drawn for the three chosen simple probability models,

such an analysis is far from being close to optimum or even simple when noise multiplication

(NM) is used, either with full data or with top-coded data. We have essentially relied on

the asymptotic theory, providing enough computational details of the MLEs and observed

Fisher information matrices in each case. Second, we should point out that our approach to

modify the microdata to protect the confidentiality of all records and carry out the analysis

based on noise-modified microdata data is different from modifying the microdata when the

goal is to release tables (Evans, Slanta, and Zayatz, 1998). Moreover, the focus of this paper

is on data analysis methods based on noise-modified microdata rather than on a study of

the effectiveness of the procedures in protecting the data.

2. DATA ANALYSIS UNDER FULL NOISE MULTIPLICATION

The problem of statistical data analysis under full noise perturbation, although old (see

Hwang, 1986; Little, 1993; Kim and Winkler, 2003), has been recently revisited, and some

new and interesting results have emerged in a nonparametric setup for estimation of the

moments and for inference about the quantiles of a variable Y based on noise multiplied

data (Nayak, Sinha and Zayatz, 2011; Sinha, Nayak and Zayatz, 2011). Briefly, Nayak et

al. (2011) discussed at length various issues related to the statistical properties of random

6

noise perturbation methods for data masking. Under the noise multiplication scenario, issues

such as confidentiality protection, moment estimation, properties of balanced noise distribu-

tion, and effects on data quality and privacy protection in the context of tabular data were

addressed at length. In a subsequent paper, Sinha et al. (2011) proposed some inferential

procedures for quantile estimation based on noise multiplied micro data. It turns out that

this is indeed a difficult inferential problem, and an empirical Bayes solution based on a

nonparametric model was developed by the authors.

As far as we know, efficient model-based parametric inference procedures based on noise

perturbed data are still lacking, with the sole exception of the work by Little (1993) where

inference under noise multiplication is briefly mentioned, with a remark that ...this approach

does not yield valid inferences for parameters..... Denoting by y1, . . . , yn a random sample of

size n from the distribution of Y ∼ fθ(y), with unknown parameter θ, and by R ∼ h(r) the

underlying noise variable (with a completely known distribution h(r)), the noise perturbed

data z = (z1, . . . , zn) with zi = yi×ri can be thought of as a random sample from Z ∼ gθ(z) =∫fθ(

zr)h(r)dr

r. The noise variables r1, . . . , rn are independent and identically distributed

(iid) as R, and it is assumed that R is a nonnegative random variable. We assume that

both Y and R are continuous random variables. When the parameters θ have moment-type

interpretations based on Y , they admit simple unbiased (not necessarily optimum) estimates

based on z (see Hwang, 1986; Nayak et al., 2011); however, efficient inference for θ based on

z is far from being simple due mainly to the possible complexity of gθ(z). In the context of

masking by noise multiplication, unlike our setup where R has a specified noise distribution,

independent of the data Y , Kim (1986), Sullivan and Fuller (1989, 1990) and Little (1993)

dealt with the case when R is made data-dependent! This procedure, while it keeps intact

certain basic moments of the original data, obviously renders considerable difficulty in the

inference process. In this context, it is rather interesting to quote Little (1993), which

indeed provides a compelling motivation for our research: Although a full likelihood-based

7

analysis may not be feasible in many settings, I think the modeling perspective provides a

useful basis for assessing simpler approximate methods. Future work might provide more

detailed applications of the modeling approach to specific masking procedures. Our goal here

is to provide exact and approximate efficient inference procedures when noise multiplication

is used as a data masking mechanism.

Here is a brief outline of our approach. In view of the anticipated complexity of the

marginal likelihood based on z, we can apply the familiar EM algorithm to compute the MLE

of θ (Little and Rubin, 2002). It is also possible to use a parametric bootstrap procedure

to study the frequentist properties of the MLE (Efron and Tibshirani, 1994). However we

instead derive the observed Fisher information matrices and use them to estimate standard

deviation in our extensive simulations. We construct 95% confidence intervals as (MLE)

± (1.96 × estimated standard deviation). Under the EM framework, we can write uobs =

(z1, ...., zn), umis = (r1, ...., rn), uc = (uobs,umis), to denote the observed data, missing data

and complete data, respectively. Using the notations in the previous paragraph, the complete

data likelihood can obviously be expressed as L(θ|uc) =∏n

i=1[fθ(ziri

)h(ri)ri

] and the observed

data likelihood as L(θ|uobs) =∏n

i=1[∫fθ(

ziri

)h(ri)ridri].

Taking logarithm, if `(θ|uc) = lnL(θ|uc) (ignoring constants), the E-step is then carried

out by starting with the estimate θ(t) (at the tth step) and computing

Q(θ|θ(t)) = Eθ(t) [`(θ|uc)|uobs] = Eθ(t)

[n∑i=1

ln fθ(ziri

)|z1, ...., zn

]=

n∑i=1

Eθ(t)

[ln fθ(

ziri

)|zi]

(1)

and the M -step is carried out by maximizing Q(θ|θ(t)) with respect to θ, resulting in θ(t+1). It

would be rather easy to evaluate the one dimensional integral (with respect to r) in Q(θ|θ(t))

above, either explicitly or numerically. The iteration defined through the E and M -steps

can then be run until a stopping criterion is met. For many choices of fθ(y), the M-step will

have a closed form expression. The E-step is also quite feasible since the one-dimensional

8

integrals appearing in (1) are straightforward to evaluate using numerical or Monte Carlo

methods.

As for the noise distribution, we have taken a popular choice R ∼ h(r) ∼ Uniform(1 −

ε, 1 + ε) for some 0 < ε < 1 with E(R) = 1 and var(R) = σ2r = ε2/3. Computation of the

MLEs, and the derivation of the observed Fisher information are both described in Appendix

B of the supplementary material for the exponential, normal and lognormal models. For

exponential and lognormal distributions, we have also used what we call customized noise

distributions that can provide simplified inference because they permit closed form evaluation

of gθ(z). These are given by

Under the exponential distribution : R ∼ hδ(r) =δδ+1

Γ(δ + 1)r−δ−2e−

δr , δ > 1,

Under the lognormal distribution : R ∼ lognormal with lnR ∼ N(−ψ2

2, ψ2).

3. DATA ANALYSIS UNDER NOISE MULTIPLICATION OF EXTREME VALUES

We now consider the set up where top coding is used for disclosure control. The random

variables yi, ri, and zi, i = 1, ...., n, are as defined in Section 2. Now, any observation yi

which exceeds a specified threshold C > 0 is considered sensitive and, under top coding such

values are simply not reported. Another option is to report zi, the noise perturbed version

of yi, along with an indicator that the true value has been perturbed. More precisely, for

i = 1, ...., n, let us define

∆i = I(yi ≤ C), and xi =

yi, if yi ≤ C,

zi, if yi > C.(2)

Inference for θ will be based on {(x1,∆1), ...., (xn,∆n)}, or based on {x1, ...., xn}. We note

that the latter data do not directly identify which observations have been perturbed, and

9

hence provide more disclosure control than the former. The expression for the joint distri-

bution of (xi, ri,∆i), which is crucial for our purpose, is given by the following proposition,

whose proof appears in Appendix A.

Proposition 1. The joint pdf of (xi, ri,∆i) is given by

kθ(xi, ri, δi) =

fθ(xi)h(ri), if xi < C, 0 < ri <∞, δi = 1,

fθ(xi/ri)h(ri)(1/ri), if 0 < ri < xi/C, δi = 0,

0, if δi = 0, ri > xi/C,

or if δi = 1, xi > C.

(3)

Proposition 1 can be used to derive the likelihood function of θ based on data of the two

types: {(x1,∆1), ...., (xn,∆n)} and (x1, ...., xn); details appear in Appendix A. We refer to

the former data type as case (i), and the latter as case (ii). Letting kθ(x, δ) and kθ(x) be the

densities defined in Appendix A, we now describe details of the computation of the maximum

likelihood estimate of θ based on the EM algorithm. We shall separately consider cases (i)

and (ii).

Case (i): Observed data consist of {(x1,∆1), ...., (xn,∆n)}.

In order to describe the EM algorithm under this setting, we define the observed, miss-

ing, and complete data, respectively, as follows: vi,obs = (x1, ...., xn,∆1, ....,∆n), vi,mis =

(r1, ...., rn), vc = (vi,obs,vi,mis). By Proposition 1, the joint density function of vc is∏ni=1[fθ(xi)h(ri)]

∆i [fθ(xi/ri)h(ri)(1/ri)]1−∆i , and thus we define the complete data likelihood

function as L(θ|vc) =∏n

i=1 fθ(xi)∆ifθ(xi/ri)

1−∆i , and the complete data log-likelihood func-

tion as

`(θ|vc) =n∑i=1

{∆i ln f(xi; θ) + (1−∆i) ln fθ(xi/ri)} . (4)

10

The E and M -steps are then computed as follows.

E-step. To carry out the E-step, we compute:

Q(θ|θ(t)) = Eθ(t) [`(θ|vc)|vi,obs]

=n∑i=1

{∆i ln fθ(xi) + Eθ(t) [(1−∆i) ln fθ(xi/ri)|xi,∆i]}

=n∑i=1

{∆i ln fθ(xi) + (1−∆i)Eθ(t) [ln fθ(xi/ri)|xi,∆i = 0]}

=n∑i=1

{∆i ln fθ(xi) + (1−∆i)ψ(xi, θ

(t), θ)}, (5)

where

ψ(xi, θ(t), θ) ≡ Eθ(t) [ln fθ(xi/ri)|xi,∆i = 0]

=

∫ ∞0

[ln fθ(xi/ri)]kθ(t)(xi, ri,∆i = 0)

kθ(t)(xi,∆i = 0)dri

=

∫ xi/C0

[ln fθ(xi/ri)]fθ(t)(xi/ri)h(ri)r−1i dri∫ xi/C

0fθ(t)(xi/ω)h(ω)ω−1dω

. (6)

M-step. If θ(t) is the current estimate of θ, then the M step updates the current estimate to

θ(t+1) = arg maxθQ(θ|θ(t)).

Case (ii): Observed data consist of only (x1, ...., xn).

To describe the EM algorithm in this setting, we re-define the observed and missing

data, respectively, as vii,obs = (x1, ...., xn), vii,mis = (r1, ...., rn,∆1, ....,∆n). The complete

data vc = (vii,obs,vii,mis) remains the same as in case (i), and hence the complete data

log-likelihood function is given by (4). The E and M -steps are then modified as follows.

11

E-step. To carry out the E-step, we now compute:

Q(θ|θ(t)) = Eθ(t) [`(θ|vc)|vii,obs]

=n∑i=1

{Eθ(t) [∆i|xi] ln fθ(xi) + Eθ(t) [(1−∆i) ln fθ(xi/ri)|xi]}

=n∑i=1

{ψ1(xi, θ

(t)) ln fθ(xi) + ψ2(xi, θ(t), θ)

}, (7)

where

ψ1(xi, θ(t)) = Eθ(t) [∆i|xi] = Prθ(t) [∆i = 1|xi] =

kθ(t)(xi,∆i = 1)

kθ(t)(xi)

=I(xi < C)fθ(t)(xi)

I(xi < C)fθ(t)(xi) + I(xi > 0)∫ xi/C


, (8)

and

ψ2(xi, θ(t), θ) = Eθ(t) [(1−∆i) ln fθ(xi/ri)|xi]

=1∑

∆i=0

∫ ∞0

(1−∆i)[ln fθ(xi/ri)]kθ(t)(xi, ri, δi)

kθ(t)(xi)dri

=

∫ ∞0

[ln fθ(xi/ri)]kθ(t)(xi, ri, δi = 0)

kθ(t)(xi)dri

=

∫ xi/C0

[ln fθ(xi/ri)]fθ(t)(xi/ri)h(ri)r−1i dri

I(xi < C)fθ(t)(xi) + I(xi > 0)∫ xi/C


. (9)

M-step. As usual, if θ(t) is the current estimate of θ, then the M step updates the current

estimate to θ(t+1) = arg maxθQ(θ|θ(t)).

Application of the above computational steps to the exponential, normal and lognormal

models is explained in Appendix C of the supplementary material, and expression for the

observed Fisher information are provided.

12

4. DESCRIPTION OF MULTIPLE IMPUTATION METHODS

In the context of statistical disclosure control, MI is used to create public use data that

protect confidential records from disclosure, while still yielding valid inferences for population

quantities. The MI methodology is generally motivated from a Bayesian perspective, but it

has been shown that the inferences obtained are often properly calibrated in the frequentist

sense; we refer to the references given above. To implement this method, the data collector,

who would have access to the confidential records, formulates an appropriate model for the

data. To complete the Bayesian model specification, noninformative prior distributions are

usually placed on model parameters. A public use data file is then created by deleting all

confidential records (which may include all or part of the data) and imputing the deleted

values, say m > 1 times, using m independent draws from the posterior predictive distribu-

tion. Hence, the public use file appears as m data sets, each of the same structure as the

original, but with confidential records replaced by imputed values. Such a public use file is

referred to in the literature as a synthetic data file. The number of imputations m is chosen

to be larger than 1, often in the range of 5-10, which allows the additional variance due

to imputation to be properly incorporated into the inferences obtained from the synthetic

data. The above is a brief general description of MI for statistical disclosure control which

is based on the method of Reiter (2003). Variants of this method have also appeared in the

literature, appropriate for certain types of data; these include imputing the population m

times, and then sampling from each synthetic population as in Raghunathan, Reiter, and

Rubin (2003).

We now briefly review the MI methods which are applicable to the two general scenarios

considered in this paper: (1) the case of total MI, which serves as a competitor to the noise

perturbation methods developed in Section 2, and (2) the case of MI for extreme values only,

which serves as a competitor to the noise perturbation methods developed in Section 3. The

13

original data are assumed to consist of an iid sample y = (y1, ...., yn) from a parametric model

fθ(y), and we place a noninformative prior distribution p(θ) on the unknown parameter θ.

We let Q = Q(θ) denote the population quantity that we wish to draw inference upon, using

the multiply imputed data. For ease of presentation, we assume Q is a scalar. Inferences

for Q are obtained from the synthetic data as follows. Let q = q(y) denote an estimator

of Q that is computed from the original data, and let v = v(y) denote an estimator of the

variance of q, also computed from the original data. For instance, q may be the maximum

likelihood estimator of Q, and v may be the estimated asymptotic variance obtained from

the inverse of the observed Fisher information matrix. Let y∗(1), ....,y∗(m) denote the m sets

of multiply imputed/synthetic data. We discuss how to generate these multiply imputed

data sets in the subsections below. Given the synthetic data, one then proceeds to compute

qj = q(y∗(j)) and vj = v(y∗(j)), the analogs of q and v, on the jth synthetic data set, for

j = 1, ....,m. The appropriate MI estimators for the settings considered here are given by

Reiter (2003). The MI estimator of Q is qm = 1m

∑mj=1 qj, and the estimator of the variance

of qm is Tm = bm/m + vm, where bm = 1m−1

∑mj=1(qj − qm)2 and vm = 1

m

∑mj=1 vj. We next

discuss the generation of y∗(1), ....,y∗(m) for the cases of total MI, and MI for extreme values.

4.1 Total MI

In the case of total MI, the entire original data set is confidential. Thus, as a method

of statistical disclosure control, total MI serves as an alternative to the noise multiplication

methods developed in Section 2. The posterior distribution is obtained as usual via Bayes

theorem: p(θ|y) ∝ p(θ)×∏n

i=1 fθ(yi). The synthetic data are then obtained as follows.

1. Draw θ∗ from the posterior distribution p(θ|y).

2. Draw y∗ = (y∗1, ...., y∗n) as iid from fθ∗(y).

Steps (1) and (2) above are repeated independently m times to obtain y∗(1), ....,y∗(m), the

14

multiply imputed/synthetic data. The notation fθ∗(y) denotes the pdf fθ(y) with the un-

known parameter θ set equal to θ∗.

4.2 MI for Extreme Values

In the case of MI for extreme values, only those data values above a known threshold

C are considered to be confidential. Thus, as a method of statistical disclosure control, MI

for extreme values serves as an alternative to the noise multiplication methods developed in

Section 3. In this scenario, we consider two MI methods presented by An and Little (2007),

namely, parametric MI based on complete data (PMIC) and parametric MI based on deleted

data (PMID). We now briefly describe these methods. Although it is only the values above

C that are considered sensitive, a cut-point CI < C is selected, and any value yi > CI is

imputed. As discussed by An and Little (2007), by choosing the cut-point CI to be less than

C, a mixing of sensitive and non-sensitive values is achieved, which should enhance the level

of protection against disclosure. Let y = (yret,ydel) where yret = {yi : yi ≤ CI} denotes the

values in y that will be retained, and ydel = {yi : yi > CI} denotes the values in y that will

be deleted.

The PMIC method proceeds as follows. One first fits the parametric model to the com-

plete data y, and thus obtains the posterior distribution p(θ|y) ∝ p(θ)×∏n

i=1 fθ(yi). Then

the synthetic data are generated as follows.

1. Draw θ∗ from the posterior distribution p(θ|y).

2. Obtain the imputed data y∗del, which is of the same length as ydel, by taking iid draws

from the truncated version of fθ(y) defined as f(CI ,∞)(y|θ∗) =fθ∗ (y)×I(CI,∞)(y)∫∞

CIfθ∗ (u)du

.

Steps (1) and (2) above are repeated independentlym times to obtain y∗(1)del , ....,y

∗(m)del . Finally,

we obtain the synthetic data as y∗(j) = (yret,y∗(j)del ), j = 1, ....,m.

Now we describe the PMID method. In the PMID method, one fits the parametric model

to the deleted data ydel instead of the complete data. That is, the posterior distribution of θ

15

is computed as p(θ|ydel) ∝ p(θ)×∏{i:yi∈ydel} fθ(yi), and synthetic data are generated through

the following steps.

1. Draw θ∗ from the posterior distribution p(θ|ydel).

2. Obtain the imputed data y∗del, which again is of the same length as ydel, by taking iid

draws from fθ∗(y). Note here that we do not draw from the truncated distribution as

in PMIC.

As usual, steps (1) and (2) above are repeated independently m times to get y∗(1)del , ....,y

∗(m)del ,

and the synthetic data are then obtained as y∗(j) = (yret,y∗(j)del ), j = 1, ....,m. A comparison

of PMID and PMIC methods will be reported later, based on numerical results.

4.3 Details for the Exponential, Normal and Lognormal Distributions

We now briefly review the forms of the prior and posterior distributions under the ex-

ponential, normal, and lognormal models. In the following, we present the form of the

posterior assuming that the complete data y are used to fit the model. We also use the

notation y to denote the observed data. The necessary adjustments for the PMID method

are straightforward.

Exponential distribution. In this case we have fθ(y) = 1θe−y/θ, 0 < y < ∞, and we take

the improper uniform prior p(θ) ∝ 1, 0 < θ < ∞. The posterior distribution is readily

obtained as

p(θ|y) =(∑n

i=1 yi)n−1

Γ(n− 1)θ−(n−1)−1e−(

∑ni=1 yi)/θ, 0 < θ <∞,

which has the form of an inverse gamma distribution. That is, (θ−1|y) ∼ Gamma(n− 1, 1∑n

i=1 yi

).

Normal distribution. In this case we have fθ(y) = 1σ√

2πe−(y−µ)2/(2σ2), −∞ < y <∞, θ =

(µ, σ2), and we specify a standard noninformative prior which places a uniform distribution

on (µ, lnσ) and assumes these two quantities to be independent a priori. Equivalently, the

16

prior on (µ, σ2) is p(µ, σ2) ∝ (σ2)−1,−∞ < µ <∞, 0 < σ2 <∞. The posterior distribution

is then readily obtained as p(µ, σ2|y) = p(µ|σ2,y)p(σ2|y) where

(σ2|y) ∼ (n− 1)s2

χ2n−1

, (µ|σ2,y) ∼ N(y, σ2/n),

with y = 1n

∑ni=1 yi and s2 = 1

n−1

∑ni=1(yi − y)2 (Gelman et al., 2003).

Lognormal distribution. In this case we have fθ(y) = 1yσ√

2πe−(ln y−µ)2/(2σ2), 0 < y < ∞,

θ = (µ, σ2). But since ln yi ∼ N(µ, σ2), we simply apply the normal distribution results

upon replacing yi by ln yi, CI by lnCI , and C by lnC.

5. ANALYSIS OF TOP CODED DATA

The statistical analysis of top coded data is essentially asymptotic in nature, based on the

MLEs of the parameters and the observed Fisher information. The MLE computation along

with the expressions for the observed Fisher information matrix for the three parametric

models are explained in Appendix D of the supplementary material.

6. NUMERICAL RESULTS

In order to compare multiple imputation with noise multiplication in terms of accuracy

of the inference, we performed a rather extensive simulation study in the context of the one

parameter exponential distribution, the normal distribution, and the lognormal distribution.

Furthermore, under each distribution, we considered the case of full noise perturbation and

total multiple imputation, as well as the case of noise perturbation and multiple imputation

for the situation of top coded data. Tables resulting from the simulation study are provided

in Appendix E of the supplementary material.

17

The setups considered for the simulation are as follows. For the exponential distribution,

the mean was chosen to be one, and the inference problem considered is the point and

interval estimation of the mean. For the normal and lognormal distributions, simulations

were done under the parameter choices µ = 0 and σ2 = 1. For the normal distribution,

point and interval estimation were considered for the mean, variance, 84th percentile (i.e.,

µ+σ) and the 95th percentile (i.e., µ+1.645σ). For the lognormal distribution, inferences on

the same parameters, namely, µ, σ2, the 84th percentile (i.e., eµ+σ) and the 95th percentile

(i.e., eµ+1.645σ) were considered as well as those for the lognormal mean (eµ+σ2

2 ) and the

lognormal variance (e2µ+2σ2 − e2µ+σ2). A nominal confidence level of 95% was used in all

interval estimation problems. For the point estimates of the various parameters, the following

quantities were estimated by Monte Carlo simulation based on 5000 iterations: the root

mean squared error (RMSE), bias, standard deviation (SD), and expected value of estimated

standard deviation (SD). For the confidence interval, the coverage probability was estimated

and the expected length of the nominal 95% two-sided confidence interval was calculated

relative to that based on the complete data (unmasked). To facilitate the comparison, all

results shown for unmasked data are based on MLEs, observed Fisher information, and

confidence intervals of the form (MLE) ± (1.96 × estimated standard deviation). For top-

coding two choices were made for the threshold C: the 90th and 95th percentiles of each

distribution.

We shall now explain the notations used in the tables regarding the multiple imputation

and noise multiplication scenarios used in the simulations. The notation CD denotes the case

of complete data without any masking. MI denotes multiple imputation used to generate

fully masked data. MI.C and MI.D are used to denote parametric multiple imputation based

on the complete data, and based on the deleted data, respectively. These are explained in

Section 4.2. In the case of a top coded sample with a top coding threshold C, values beyond

a threshold CI were also replaced with imputed values, where CI < C, following An and

18

Little (2007). Two choices were made for CI following the recipe prescribed by An and

Little (2007). If nS denotes the number of values beyond the top coding threshold C, the

two choices of CI correspond to thresholds for which 2nS values in the sample are larger,

and 4nS values are larger. Since these values approximately correspond to the 90th and 80th

percentiles of the distribution when C is the 95th percentile, these are denoted (following

An and Little, 2007) by MI.C90 and MI.C80, respectively, in the case of multiple imputation

based on the complete data, and denoted by MI.D90 and MI.D80, respectively, in the case of

multiple imputation based on the deleted data. Throughout we have used 50 sets of imputed

values under MI.

The notations used under noise multiplication are as follows. We have used either a

Uniform(1− ε, 1+ ε) distribution or a customized distribution for generating the noise multi-

pliers. In the case of Uniform(1− ε, 1 + ε), the notation NM10U denotes noise multiplication

with the above uniform distribution under the choice ε = 0.10. NM20U, NM30U etc. are

similarly defined. Furthermore, NM10C, NM20C etc. are the corresponding notations under

the customized prior whose mean and variance match those of the respective uniform distri-

butions. Additional notation used under the uniform noise distribution Uniform(1− ε, 1 + ε)

is as follows. For the choice ε = 0.1, NM10.1 and NM10.2, respectively, represent the case of

observing the (xi,∆i)s, or only the xis; these are Case (i) and Case (ii) explained in Section 3

for the three different distributions. NM20.1, NM20.2 etc. are similarly defined. In addition

to the above notations, entries in the tables corresponding to CENS denote results from the

analysis of the censored data, i.e., the top coded data.

The statistical computing software R (R Development Core Team, 2011) was used for all

computations and the integrate function in R was used to evaluate the required univariate

integrals that could not be obtained in closed form. We shall now summarize the conclusions

emerging from the extensive numerical results with detailed tables appearing in Appendix

E of the supplementary material.

19

The Exponential Distribution.

For the case of fully masked data, confidence interval coverage probabilities resulting

from noise multiplication are not satisfactory for small sample sizes; see the entries in the

tables (supplementary material) corresponding to the sample size of 30. However, asymp-

totic inference based on the unmasked data also yields unsatisfactory coverage probability

when n = 30. Interestingly, even in this case where the sample is as small as n = 30,

multiple imputation does provide satisfactory coverage probability. The numerical results

corresponding to sample size of 50 or more show that the coverages are quite satisfactory

under the various noise multiplication scenarios, and under multiple imputation. As the

noise variance increases, the relative length increases, as expected. The RMSE values, SD,

and relative length each appear to show an increasing trend as the noise variance goes up,

but an increasing trend in the bias is not always present in the numerical results. In terms

of RMSE and relative length of confidence intervals, the noise multiplication yields results

similar to MI when ε is about 0.3 or 0.4. For smaller (larger) values of ε, the noise multi-

plication generally yields smaller (larger) RMSE and narrower (wider) confidence intervals.

In terms of bias, a much larger bias has been noted under multiple imputation, compared

to all scenarios of noise multiplication. We note that the uniform and customized noise dis-

tributions generally give similar results, with the customized distribution providing slightly

more accurate inference than uniform when ε is large.

In the case of multiple imputation and noise multiplication for top coded data, the

performance of MI.C80 and MI.C90 are not satisfactory, even for samples as large as 2000, as

the true confidence interval coverage is below the nominal level. However, both MI.D80 and

MI.D90 are satisfactory for various sample sizes, even for sample size as small as 100. All the

noise multiplication scenarios provide satisfactory coverage probabilities. The entries under

CENS show that the top coding method itself provides satisfactory coverage probabilities,

but it tends to be less accurate than the noise multiplication methods unless ε is quite large.

20

Interestingly, for n = 100 and n = 200, the top coded data and noise multiplied data (even

with large ε = 0.90) yield more accurate inferences than MI.D80 and MI.D90 in terms of

RMSE and confidence interval length. For n = 1000 and n = 2000 the noise multiplied data

are generally more or equally accurate than MI.D80 and MI.D90 for ε ≤ 0.5 or 0.6. In our

numerical results, the CENS method (top code data) never provide more accurate inference

(in terms of RMSE and confidence interval length) than the case (i) noise multiplied sample,

but they do sometimes provide more accurate inference than a case (ii) noise multiplied

sample when ε is large. At the end of this section, we explain this finding, which holds

for all three distributions considered. As one would expect, the case (i) noise multiplied

sample (xi,∆i), i = 1, . . . , n, provides greater accuracy of inference than the case (ii) noise

multiplied sample x1, . . . , xn. Regarding MI.D80 versus MI.D90, with the larger sample sizes

n = 1000 or 2000 the two methods are mostly similar. Interestingly, in the smaller sample

sizes n = 100 or 200, the MI.D90 method, which imputes less data than MI.D80, but also

uses less data to fit the imputation model, is less accurate than the MI.D80 method. The

remaining conclusions are similar to those noted above for the case of fully masked data; a

rather large bias has been noted under multiple imputation.

The Normal Distribution

Under full masking of the data, for estimating the normal mean, the results are compa-

rable under multiple imputation and noise multiplication when ε is equal to about 0.4 or

0.5 (depending on the sample size). By choosing ε less than or greater than 0.4 or 0.5, the

results under noise multiplication can be made more or less accurate than multiple impu-

tation, respectively. However, for estimating the variance under the smaller sample sizes of

n = 30, 50, 100, multiple imputation appears to have an edge in terms of coverage probabil-

ity and relative length, and in terms of the other quantities. For estimating the variance,

the asymptotic confidence intervals based on unmasked data and noise multiplied data all

21

have coverage probability below nominal, while confidence intervals based on multiply im-

puted data have closer to nominal coverage probability. This effect diminishes for both the

unmasked and noise multiplied data for the larger sample sizes n = 200, 1000, 2000. For

estimating the percentiles, the results are mostly similar between multiple imputation and

noise multiplication and one can locate a value of ε (e.g., ε = 0.2 or 0.3 here) at which the

inferences have nearly equivalent accuracy.

Under noise multiplication or multiple imputation for top coded data, MI.C90 and

MI.C80 again have poor coverages even with large samples. With the smaller sample size of

n = 100 we notice that when the parameter of interest is the variance σ2 or µ + σ2/2 the

MI.D90 method can yield a huge RMSE, bias, SD, and relative confidence interval length,

while the coverage probability is maintained at the nominal level. Such inaccurate inference

occurs under MI.D90 because when n = 100 and C is the 95th percentile, there is a non-

negligible probability that the number of data points used to fit the imputation model can be

as small as 2 if nS = 1 (refer to Section 4.2); and when this happens there is a non-negligible

probability that the posterior draw for σ2 is huge. Apart from this anomaly in MI.D90, the

results between MI.D90, MI.D80 and the noise multiplication methods are generally similar

with the MI.D methods often yielding a noticeably high level of accuracy in terms of RMSE,

SD, and relative confidence interval length. It is interesting that in the normal case the MI.D

methods generally yield highly accurate inference, whereas in the exponential case, we noted

that their accuracy was somewhat low, sometimes lower than top coding. In each case one

can determine an ε at which the noise multiplied data provide a similar level of accuracy

as MI. For instance when estimating the normal mean, a value of ε = 0.4 generally gives a

similar level of accuracy as the MI.D methods.

The Lognormal Distribution

For fully masked data, when estimating the log scale mean µ, the results are very much in

22

line with those noted above for the normal case for estimation of the normal mean. Similarly,

when estimating the log scale variance σ2, the results are in line with the findings noted above

for the normal case for estimation of the normal variance. With the smaller sample sizes of

n = 30, 50, 100, the noise multiplied data and unmasked data also provide less than nominal

confidence interval coverage when the parameter of interest is the lognormal mean, lognormal

variance, the 84th percentile and the 95th percentile. Multiple imputation provides closer to

nominal coverage probability in these cases. For the larger sample sizes n = 200, 1000, 2000,

this effect diminishes. Regarding the choice of uniform vs. customized noise distributions,

the findings are similar to those of the exponential case.

For noise multiplication or multiple imputation for top coded data, we note that MI.C90

and MI.C80, exhibit poor confidence interval coverages, similar to what has been noted

earlier. When the parameter of interest is the variance, lognormal mean, lognormal variance,

84th percentile, or 95th percentile, and when n = 100 or 200, the MI.D80 and MI.D90

methods can yield huge RMSE, bias, SD and relative confidence interval length. This effect

occurs for reasons similar to those noted above for the normal case, namely, there is a non-

negligible probability that the number of data points available to fit the model is as small as

four (MI.D80) or two (MI.D90), i.e., nS = 1. In these cases the noise multiplication methods

can still provide reasonable results. In larger sample sizes, the multiple imputation and noise

multiplication methods both provide reasonably accurate results, and one can select an ε at

which their accuracy is nearly equivalent (in some cases such as when estimating the log

scale mean, the ε value can be fairly large).

Remarks on Noise Multiplication Versus Top Coding at C

1. If we observe noise multiplied data under case (i) of Section 3, then a top coded

sample can be re-constructed from the observed data. However, a case (i) noise multiplied

sample cannot be constructed from the top coded data. So in this sense, case (i) noise

23

multiplied data carry more information than top coded data. The simulation results confirm

this statement because the case (i) noise multiplication method generally leads to a shorter

confidence interval and smaller SD than top coding for all values of ε that we consider. As

ε increases, the case (i) noise multiplication method begins to yield results which are quite

similar to top coding, as one would expect.

2. Suppose we observe noise multiplied data under case (ii) of Section 3 and the noise

density is Uniform(1− ε, 1 + ε). Then the top coded sample cannot be re-constructed from

these observed data because for any xi such that (1 − ε) × C < xi < C, we do not know

whether or not this xi was noise perturbed. Of course, we also cannot re-construct a case

(ii) noise multiplied sample based on only the top coded data. Therefore, depending on the

value of ε, the case (ii) noise multiplied sample may contain either more or less information

than the top coded sample. The simulation results confirm this statement since for small ε,

the case (ii) noise multiplication scenario leads to a shorter confidence interval and smaller

SD than top coding. When ε becomes large, this situation is reversed. It is interesting to

note that given the noise multiplied data under case (ii), we could construct a top coded

sample with a lower top code value of (1− ε)× C.

7. DATA ANALYSIS

In this section we provide an application based on data from the 2000 U.S. Current

Population Survey which has been thoroughly and repeatedly analyzed in Reiter (2005c,d;

2009) and Drechsler and Reiter (2010) for illustrating various aspects of multiple imputation

methodology. Here we shall use the data on two variables whose distributions can be well

approximated by the lognormal distribution. We shall report the analysis based on full noise

multiplication and total MI (Tables F1 and F2 in Appendix F of the supplementary material).

We shall also top code 10% of the data and then report the analysis based on the full data

24

sets, the censored data sets resulting from top coding, the data sets obtained by multiple

imputation of the top coded values, and the data sets resulting from noise multiplication of

the top coded values (Tables 1 and 2). This will of course permit us to compare the results

of the different analysis for the two data sets.

The entire data comprise of seventeen variables measured on 51,016 heads of households

(see Table 1 in Drechsler and Reiter, 2010 for a description of nine variables) including age,

race, sex and marital status as key identifiers, and a mix of other categorical and numerical

variables with non-Gaussian distributions (with large percentages of values equal to zero).

We have selected two such numerical variables with extreme sample sizes, namely, total

household income with sample size = 50,661 and household alimony payment with sample

size 206, deleting the 0 values, and have carried out the analysis. It turns out that a lognormal

distribution fits both the datasets well.

Results of the analysis are given in the Tables 1 − 2 (and Tables F1 − F2 of the supple-

mentary material). As in the previous section, we used the statistical computing software R

(R Development Core Team, 2011) for all computations, and 50 multiple imputations were

used in the MI methods. The notations used in the tables are as explained in Section 6. In

addition, Est denotes point estimate of the parameter and Rel. Len. denotes the length of

a 95% confidence interval, relative to that of the interval based on the complete data. We

note that as with simulated data the conclusions about the common parameters of interest

are similar based on multiple imputation and noise perturbation (with noise variance not

exceeding about 8.33%, ε = 0.5 in uniform case) and both almost reproduce the estimates

based on the original unmasked data. It is interesting to note from Table 1 (total household

income) that the MI.C methods yield considerably wider confidence intervals than the MI.D

and noise multiplication methods with small to moderate ε. In these cases the MI.C methods

also yield larger point estimates of the lognormal mean, lognormal variance, 84th percentile

and 95th percentile in comparison with the other methods. The results of Table 2 are based

25

on a much smaller sample size and here we do not observe such wide confidence intervals

under the MI.C methods.

8. DISCUSSION

How to alter the data before releasing it to the public continues to be a vital concern

of statistical agencies in order to minimize the risk of disclosure. However, this goal has to

be balanced with the interest of preserving the utility of the released data. A number of

strategies exist for achieving the goal of disclosure limitation, and for accurately analyzing the

released data. Perhaps not much effort has been devoted to compare the different disclosure

limitation strategies in terms of accuracy of the inference resulting from the released data.

If a particular data masking approach can provide more accurate inference compared to

other competing approaches, this should be of obvious interest to statistical agencies, survey

organizers, and the public. This work is an attempt in this direction, and our goal has been

two-fold: (a) to develop rigorous data analysis methodologies for noise multiplied data under

noise multiplication of the entire data, or under noise multiplication of the top coded data,

and (b) to compare noise multiplication and multiple imputation in terms of accuracy of

the resulting inference. We have developed all the necessary theoretical results for analyzing

noise multiplied data, when the original (unmasked) sample arises from three parametric

distributions: exponential, normal and lognormal distributions. Masking the entire data, and

masking data above a threshold, are treated separately. As is well known, the latter situation

is very common in the situation of top coding, where sample values above a threshold are

suppressed for privacy protection. We have also reported extensive numerical results to assess

the accuracy of the inference concerning a variety of parameters. The numerical results have

brought out the similarities, differences and advantages of the two data masking techniques

we have investigated.

26

In the top code scenario where only values above a threshold C > 0 require protection,

we have considered noise multiplication under two data release scenarios: case (i) in which

each released value includes an indicator of whether or not it was noise perturbed, and case

(ii) in which no such indicator is provided. We developed data analysis methods under both

cases, and argued that case (i) should always provide more (or equally) accurate inference

than top coding at C, while case (ii) can provide either more or less accurate inference than

top coding at C, depending on the magnitude of the noise variance. Our empirical results

show that both cases provide almost equally accurate inferences when the noise variance

is small, and the difference in accuracy increases, but perhaps is not too substantial, when

the noise variance is moderately large. The case (ii) data release may sometimes be more

desirable for statistical agencies than the case (i) data release, as case (ii) would appear to

provide an enhanced level of protection against disclosure. The results of this article show

how to obtain valid inferences in both cases.

In summary, parametric statistical procedures for fully masked data based on multiple

imputation and noise multiplication can provide comparable inferences in many cases that we

considered. The inferences obtained from these two different methods can usually be made

nearly equivalent by setting the noise variance appropriately. When a large noise variance

providing a high degree of confidentiality is used, inference is typically less accurate but

still reasonable. In the case of top coded data, similar conclusions hold. We note however

that the procedures MI.C90 and MI.C80 do not perform well in most cases as they tend

to produce confidence intervals whose true coverage probability is below the nominal level

even when the sample size is quite large. On the other hand, the methods MI.D90 and

MI.D80 generally do yield satisfactory results, but as noted in Section 6, for smaller sample

sizes, there is a potential for a large RMSE, SD, and wide confidence interval under these

methods. Compared to multiple imputation, an appealing feature of noise multiplication

is its flexibility; the noise variance acts as a tuning parameter, and its choice allows one

27

to balance data quality with confidentiality protection. The results of this article provide

guidance regarding the effect of noise variance on data quality. Our results show, in particular

situations, precisely the value of the noise variance at which noise multiplied data will provide

inferences with nearly equivalent accuracy as synthetic (multiply imputed) data. By setting

the noise variance smaller than this value, we have shown that one can obtain inferences that

are more accurate than synthetic data, but perhaps at the expense of a higher disclosure risk.

Similarly, one can increase the noise variance, decreasing accuracy of inferences, but perhaps

enhancing the level of protection against disclosure. Indeed the impact of the noise variance

on disclosure risk requires further investigation. While inferences become less accurate when

the noise variance is large, we note that the inferences generally are still valid, i.e., confidence

interval coverage probability is generally maintained at the nominal level and bias is small.

28

Table 1. Analysis of household total income with protection for values above C = 90th empiricalpercentile

µ σ2 eµ+σ2/2 e2µ+2σ2 − e2µ+σ2eµ+σ eµ+1.645σ

Est Rel. Est Rel. Est Rel. Est× 10−6 Rel. Est Rel. Est Rel.Len. Len. Len. Len. Len. Len.

CD 10.495 1.000 0.981 1.000 58996 1.000 5805.083 1.000 97263 1.000 184262 1.000MI.C90 10.526 1.049 1.078 1.101 63865 1.156 7903.475 1.447 105220 1.137 205534 1.173MI.C80 10.551 1.075 1.130 1.159 67245 1.262 9478.622 1.798 110648 1.229 219649 1.289MI.D90 10.494 1.000 0.981 1.001 58965 1.001 5793.529 1.000 97212 1.001 184129 1.001MI.D80 10.494 1.001 0.981 1.002 58967 1.003 5795.687 1.002 97215 1.003 184146 1.003CENS 10.510 1.040 1.037 1.140 61592 1.150 6906.667 1.338 101531 1.139 195816 1.178NM10.1 10.495 1.001 0.982 1.002 59048 1.002 5824.788 1.005 97349 1.002 184486 1.003NM10.2 10.495 1.001 0.982 1.002 59051 1.002 5826.076 1.005 97355 1.002 184501 1.003NM20.1 10.496 1.002 0.985 1.006 59169 1.008 5872.738 1.017 97551 1.007 185022 1.009NM20.2 10.496 1.003 0.985 1.007 59200 1.009 5884.083 1.020 97602 1.008 185152 1.010NM30.1 10.497 1.004 0.987 1.012 59293 1.014 5921.903 1.031 97756 1.013 185567 1.017NM30.2 10.497 1.006 0.989 1.015 59384 1.018 5955.307 1.039 97907 1.017 185951 1.021NM40.1 10.497 1.006 0.990 1.019 59420 1.021 5971.468 1.046 97965 1.020 186120 1.025NM40.2 10.499 1.009 0.994 1.025 59639 1.031 6052.017 1.067 98327 1.029 187037 1.036NM50.1 10.499 1.009 0.995 1.028 59650 1.033 6065.054 1.071 98345 1.030 187139 1.038NM50.2 10.502 1.016 1.002 1.042 60084 1.053 6226.125 1.114 99061 1.050 188952 1.061NM60.1 10.499 1.011 0.996 1.034 59685 1.038 6080.100 1.080 98403 1.035 187299 1.044NM60.2 10.505 1.023 1.008 1.058 60404 1.075 6349.797 1.154 99589 1.070 190308 1.085NM70.1 10.501 1.014 1.001 1.045 59942 1.050 6186.262 1.108 98828 1.047 188442 1.059NM70.2 10.510 1.035 1.021 1.086 61117 1.114 6635.825 1.238 100760 1.106 193357 1.129NM80.1 10.502 1.017 1.005 1.053 60128 1.060 6263.484 1.130 99133 1.056 189264 1.070NM80.2 10.515 1.050 1.031 1.114 61707 1.156 6871.419 1.318 101725 1.146 195836 1.175NM90.1 10.502 1.018 1.007 1.060 60221 1.066 6303.106 1.144 99286 1.061 189680 1.078NM90.2 10.518 1.070 1.039 1.141 62177 1.201 7064.943 1.396 102492 1.189 197822 1.223

29

Table 2. Analysis of household alimony payments with protection for values above C = 90thempirical percentile



CD 8.715 1.000 1.168 1.000 10930 1.000 264.812 1.000 17962 1.000 36069 1.000MI.C90 8.711 1.007 1.183 1.018 10976 1.022 277.204 1.098 18016 1.016 36370 1.025MI.C80 8.725 1.022 1.215 1.049 11337 1.085 314.648 1.320 18574 1.069 37884 1.092MI.D90 8.710 0.994 1.154 0.991 10804 0.985 255.589 0.978 17759 0.986 35529 0.984MI.D80 8.718 1.006 1.179 1.013 11038 1.026 278.497 1.096 18124 1.021 36543 1.028CENS 8.740 1.055 1.277 1.181 11829 1.224 361.734 1.574 19338 1.192 40082 1.251NM10.1 8.715 1.000 1.168 1.000 10922 0.999 264.096 0.998 17949 0.999 36034 0.999NM10.2 8.715 1.000 1.169 1.001 10934 1.001 265.245 1.003 17969 1.001 36088 1.002NM20.1 8.715 1.000 1.168 1.002 10930 1.002 264.725 1.002 17962 1.002 36066 1.002NM20.2 8.715 1.000 1.168 1.003 10930 1.002 264.783 1.003 17962 1.002 36069 1.003NM30.1 8.719 1.007 1.182 1.017 11046 1.023 275.907 1.056 18142 1.020 36581 1.025NM30.2 8.719 1.007 1.182 1.018 11042 1.023 275.624 1.056 18137 1.021 36567 1.026NM40.1 8.719 1.007 1.182 1.020 11047 1.026 276.037 1.060 18143 1.023 36586 1.029NM40.2 8.717 1.005 1.176 1.016 10990 1.019 270.739 1.038 18055 1.017 36340 1.021NM50.1 8.724 1.017 1.203 1.043 11219 1.059 293.152 1.145 18411 1.051 37349 1.065NM50.2 8.721 1.014 1.193 1.039 11130 1.048 284.737 1.111 18272 1.043 36967 1.054NM60.1 8.724 1.019 1.207 1.051 11246 1.067 296.198 1.165 18451 1.058 37474 1.075NM60.2 8.720 1.018 1.200 1.054 11165 1.064 289.284 1.145 18324 1.056 37144 1.072NM70.1 8.725 1.019 1.206 1.055 11246 1.070 296.116 1.169 18452 1.061 37472 1.078NM70.2 8.714 1.013 1.181 1.047 10985 1.045 272.534 1.080 18043 1.041 36371 1.053NM80.1 8.725 1.021 1.207 1.063 11252 1.078 296.798 1.182 18461 1.068 37501 1.087NM80.2 8.726 1.032 1.215 1.093 11315 1.117 303.461 1.249 18558 1.104 37783 1.130NM90.1 8.724 1.019 1.203 1.059 11217 1.071 293.298 1.164 18407 1.063 37348 1.080NM90.2 8.687 1.003 1.134 1.034 10449 0.999 230.225 0.922 17191 1.004 34168 1.005

30

SUPPLEMENTARY MATERIALS

The supplementary material consists of five appendices as follows.

Appendix B: MLEs and Fisher information under full noise multiplication.

Appendix C: MLEs and Fisher information under noise multiplication of extreme values.

Appendix D: MLEs and Fisher information under top coding.

Appendix E: Simulation results.

Appendix F: Analysis of fully masked data on total household income and household al-

imony payment based on data from the 2000 U.S. Current Population Survey.

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34

APPENDIX A: PROOF OF PROPOSITION 1 AND SOME RELATED RESULTS

We let Y ∼ fθ(y), independent of R ∼ h(r), as defined in Section 2 and we let ∆ = I(Y ≤ C),

X = Y if Y ≤ C and = R× Y if Y > C, as defined in Section 3, where C > 0 is a constant.

Then letting Fθ(y) =∫ y−∞ fθ(t)dt and H(r) =

∫ r0h(t)dt, we have, for r > 0,

Pr(X ≤ x,R ≤ r,∆ = 1) = Pr(X ≤ x,R ≤ r |∆ = 1) Pr(∆ = 1)

= Pr(Y ≤ x,R ≤ r |Y ≤ C)P (Y ≤ C)

= Pr(Y ≤ x, Y ≤ C,R ≤ r)

=

Fθ(x)H(r), if x ≤ C,

Fθ(C)H(r), if x > C,(10)

and, for r > 0, C > 0,

Pr(X ≤ x,R ≤ r,∆ = 0) = Pr(X ≤ x,R ≤ r |∆ = 0) Pr(∆ = 0)

= Pr(Y R ≤ x,R ≤ r |Y > C) Pr(Y > C)

= Pr(Y R ≤ x, Y > C,R ≤ r)

= Pr(C < Y ≤ x

R,R ≤ r)

=

∫ r

0

∫ x/ωC

fθ(y)h(ω)dydω, if r ≤ xC,∫ x/C

0

∫ x/ωC

fθ(y)h(ω)dydω, if r > xC> 0,

0, if x ≤ 0,

=

∫ r

0[Fθ(x/ω)− Fθ(C)]h(ω)dω, if r ≤ x

C,∫ x/C

0[Fθ(x/ω)− Fθ(C)]h(ω)dω, if 0 < x

C< r,

0, if x ≤ 0.

(11)

35

To obtain the joint pdf of (X,R,∆), we differentiate (10) and (11) with respect to x and r

to obtain:

∂2

∂x∂rPr(X ≤ x,R ≤ r,∆ = 1) =

fθ(x)h(r), if x < C,

0, if x > C,(12)

∂2

∂x∂rPr(X ≤ x,R ≤ r,∆ = 0) =

fθ(x/r)h(r)(1/r), if r < x

C,

0, if 0 < xC< r,

0, if x < 0,

(13)

which completes the proof of Proposition 1. Letting (x1,∆1), ...., (xn,∆n) be iid as (X,∆),

the following results then follow from Proposition 1.

Result 1. The joint pdf of (X,∆) is given by

kθ(x, δ) =

fθ(x), if x < C, δ = 1,

0, if xi > C, δ = 1,∫ x/C0

fθ(x/r)h(r)(1/r)dr, if x > 0, δ = 0,

0, if x < 0, δ = 0.

(14)

Result 2. The likelihood function for θ based on (x1,∆1), ...., (xn,∆n) is given by

L(θ|x1, ...., xn,∆1, ....,∆n) =n∏i=1

kθ(xi,∆i)

=n∏i=1

{[fθ(xi)]∆i × [

∫ xi/C

0

fθ(xi/r)h(r)(1/r)dr]1−∆i}. (15)

36

Result 3. The marginal pdf of X is given by

kθ(x) = fθ(x)I(x < C) +

∫ x/C

0

fθ(x/r)h(r)(1/r)drI(x > 0) (16)

=

fθ(x) +

∫ x/C0

fθ(x/r)h(r)(1/r)dr, if 0 < x < C,

fθ(x), if x < 0,∫ x/C0

fθ(x/r)h(r)(1/r)dr, if x > C.

Result 4. The likelihood function for θ based on x1, ...., xn is given by

L(θ|x1, ...., xn) =n∏i=1

kθ(xi)

=n∏i=1

{fθ(xi)I(xi < C) +

∫ xi/C

0

fθ(xi/r)h(r)(1/r)drI(xi > 0)}. (17)

37

SUPPLEMENTARY MATERIAL

A Comparison of Statistical Disclosure Control Methods:Multiple Imputation Versus Noise Multiplication

Martin Klein, Thomas Mathew, and Bimal Sinha

APPENDIX B: MLEs AND FISHER INFORMATION UNDER FULL NOISEMULTIPLICATION

B.1 Exponential distributionWe assume Y ∼ fθ(y) = 1

θe− yθ , 0 < y < ∞. Then the joint pdf g(z, r) of (Z,R), the marginal

pdf g(z) of Z and the conditional pdf g(r|z) of R, given Z = z, are given by

gθ(z, r) =1

θe−

zrθh(r)

r, gθ(z) =

∫ ∞0

1

θe−

zrθh(r)

rdr, gθ(r|z) =

e−zrθh(r)r∫∞

0 e−zwθ

h(w)w dw

, (1)

for 0 < r < ∞ and 0 < z < ∞, and hence the complete data likelihood L(θ|uc) and loglikelihood`(θ|uc) can be expressed as

L(θ|uc) ∼1

θne− 1θ

∑ni=1

ziri , `(θ|uc) = −n ln θ − 1

θ

n∑i=1

ziri. (2)

The E and M -steps for computing the MLE of θ based on z are as follows.E-step.

Q(θ|θ(t)) = Eθ(t) [`(θ|uc)|uobs] = −n ln θ − 1

θ

n∑i=1

ziE[1

ri|zi] (3)

M-step. Obviously Q(θ|θ(t)) is maximized with respect to θ at θ = 1n

∑ni=1 ziψ(θ(t), zi) where

ψ(θ(t), zi) = Eθ(t) [1ri|zi].

Thus the EM iterations are defined by

θ(t+1) =1

n

n∑i=1

ziψ(θ(t), zi). (4)

The computation of ψ(θ(t), z) for any z follows directly from the conditional pdf of r, given z, whichis given by (1).

In the special case when R ∼ Uniform(1− ε, 1 + ε), upon direct integration, ψ(θ(t), z) simplifiesto

ψuniform(θ(t), z) =

θ(t)

z

[e− z

θ(t)(1+ε) − e− z

θ(t)(1−ε)

]∫ 1+ε

1−ε e− z

wθ(t)dww

. (5)

38

Lastly, the observed Fisher information is computed by adding component-wise terms based onz1, . . . , zn, with a generic term as follows. Since the marginal pdf of z is given by (1), we get thefirst two derivatives of ln[gθ(z)] as

∂ ln[gθ(z)]

∂θ= −1

θ+

∫∞0 e−

zrθzh(r)r2θ2

dr∫∞0 e−

zrθh(r)r dr

, (6)

∂2 ln[gθ(z)]

∂θ2=

1

θ2− 2z

θ3

∫∞0 e−

zrθh(r)r2dr∫∞

0 e−zrθh(r)r dr

+z2

θ4

∫∞0 e−

zrθh(r)r3dr∫∞

0 e−zrθh(r)r dr

− z2

θ4

[∫∞0 e−

zrθh(r)r2dr∫∞

0 e−zrθh(r)r dr

]2

. (7)

A remark is now in order. Since the choice of the noise distribution mostly relies on the dataanalyst, a special case leading to simplified inferential methods warrants some attention. We referto this as a customized or conjugate noise distribution. We choose

R ∼ hδ(r) =δδ+1

Γ(δ + 1)r−δ−2e−

δr , δ > 1. (8)

It is easy to verify that E(R) = 1, as desired, and var(R) = σ2r = 1

δ−1 . We choose δ > 1 for adesirable level of noise variation. A direct computation shows that the pdf of Z = Y ×R is obtainedas

gθ(z) =δ + 1

θ× δδ+1

( zθ + δ)δ+2. (9)

This readily leads to the following likelihood based on z1, ...., zn:

L(θ|uobs) ∼ θ−n ×n∏i=1

(ziθ

+ δ)−δ−2. (10)

The maximum likelihood estimate of θ in this case can be directly computed by solving the equation:

d`(θ|uobs)

dθ= −n

θ+δ + 2

θ2

n∑i=1

zi( ziθ + δ)

= 0, (11)

which can be simplified asn∑i=1

zizi + θδ

=n

δ + 2. (12)

To compute the observed Fisher information about θ contained in the data uobs, we note that uponsimplification,

−d2`(θ|uobs)

dθ2=n(δ + 1)

θ2− δ2(δ + 2)

n∑i=1

1

(zi + θδ)2. (13)

39

B.2 Normal distribution

We assume Y ∼ fθ(y) ∼ 1σe− (y−µ)2

2σ2 , θ = (µ, σ2). Then the joint pdf g(z, r) of (Z,R), themarginal pdf g(z) of Z and the conditional pdf g(r|z) of R, given Z = z, are given by

gθ(z, r) ∼1

σe−

( zr−µ)2

2σ2h(r)

r, gθ(z) ∼

1

σ

∫ ∞0

e−( zr−µ)

2

2σ2h(r)

rdr, gθ(r|z) =

e−( zr−µ)

2

2σ2h(r)r∫∞

0 e−( zr−µ)

2

2σ2h(r)r dr

, (14)

for 0 < r < ∞ and 0 < z < ∞, and hence the complete data likelihood L(θ|uc) and loglikelihood`(θ|uc) are now expressed as

L(θ|uc) ∼1

σne− 1

2σ2

∑ni=1(

ziri−µ)2

, `(θ|uc) = −n lnσ − 1

2σ2

n∑i=1

(ziri− µ)2. (15)

The EM steps for computing the MLE of θ based on z are as follows.E-step. We compute

Q(θ|θ(t)) = Eθ(t) [`(θ|uc)|uobs],

= −n lnσ − 1

2σ2

n∑i=1

[z2i ψ2(θ(t), zi)− 2µziψ1(θ(t), zi) + µ2], (16)

where ψ1(θ(t), zi) = Eθ(t) [1ri|zi] and ψ2(θ(t), zi) = Eθ(t) [

1r2i|zi], and these two quantities can be readily

computed based on g(r|z) given in (14).M-step. Note that

∂Q(θ|θ(t))

∂µ=

1

σ2

n∑i=1

ziψ1(θ(t), zi)−nµ

σ2,

∂Q(θ|θ(t))

∂σ2= − n

2σ2+

1

2σ4

n∑i=1

[z2i ψ2(θ(t), zi)− 2µziψ1(θ(t), zi) + µ2

]. (17)

Hence Q(θ|θ(t)) is maximized at

µ =1

n

n∑i=1

ziψ1(θ(t), zi), σ2 =1

n

n∑i=1

z2i ψ2(θ(t), zi)− (µ)2. (18)

Thus the EM iterations are defined by

µ(t+1) =1

n

n∑i=1

ziψ1(θ(t), zi), [σ(t+1)]2 =1

n

n∑i=1

z2i ψ2(θ(t), zi)− [µ(t+1)]2. (19)

The observed Fisher information is computed by adding component-wise terms based on z1, ...., zn.A generic term can be obtained as follows. We note that the marginal pdf of Z is given by

gθ(z) ∼1

σ

∫ ∞0

e−( zr−µ)

2

2σ2h(r)

rdr. (20)

40

Writing the integrand above as gθ(z, r), we get

∂gθ(z)

∂µ=

∫ ∞0

gθ(z, r)[( zr − µ)

σ2]dr =

1

σ2

[∫ ∞0

gθ(z, r)z

rdr − µgθ(z)

], (21)

resulting in∂ ln gθ(z)

∂µ=

1

σ2

[∫∞0 gθ(z, r)

zrdr

gθ(z)− µg(z)

], (22)

and hence in∂2 ln gθ(z)

∂µ2=

1

σ2

∂

∂µ

[∫∞0 gθ(z, r)

zrdr

gθ(z)

]− 1

σ2. (23)

Note that

∂

∂µ

[∫∞0 gθ(z, r)

zrdr

gθ(z)

]=

∂∂µ

∫∞0 gθ(z, r)

zrdr

gθ(z)−

[∫∞

0 gθ(z, r)zrdr][

∂gθ(z)∂µ ]

g2θ(z)

. (24)

Since

∂

∂µ

∫ ∞0

gθ(z, r)z

rdr =

∫ ∞0

gθ(z, r)z

r(z

r− µ)

1

σ2dr

=z2

σ2

∫ ∞0

gθ(z, r)dr

r2− zµ

σ2

∫ ∞0

gθ(z, r)dr

r, (25)

upon using (23) and simplifying, we get

∂2 ln gθ(z)

∂µ2=

z2

σ4gθ(z)

[∫ ∞0

gθ(z, r)dr

r2−

(∫∞

0 gθ(z, r)drr )2

gθ(z)

]− 1

σ2. (26)

We next compute ∂gθ(z)∂σ2 which easily simplifies to

∂gθ(z)

∂σ2= − 1

2σ2gθ(z) +

1

2σ4

∫ ∞0

gθ(z, r)(z

r− µ)2dr, (27)

and readily leads to∂ ln gθ(z)

∂σ2=

1

2σ4

∫∞0 gθ(z, r)(

zr − µ)2dr

gθ(z)− 1

2σ2. (28)

Keeping aside the last term, the partial derivative of the first term again with respect to σ2 isobtained as

∂

∂σ2

[1

2σ4

∫∞0 gθ(z, r)(

zr − µ)2dr

gθ(z)

]= − 1

σ6

∫∞0 gθ(z, r)(

zr − µ)2dr

gθ(z)

+1

2σ4

∫∞0

∂∂σ2 gθ(z, r)(

zr − µ)2dr

gθ(z)

− 1

2σ4

∫∞0 gθ(z, r)(

zr − µ)2dr × ∂

∂σ2 gθ(z)

g2θ(z)

. (29)

41

Using the expressions for ∂∂σ2 gθ(z, r) and ∂

∂σ2 gθ(z) given above in (27) and simplifying, we get

∂2

(∂σ2)2ln gθ(z) = − 1

σ6

∫∞0 gθ(z, r)(

zr − µ)2dr

gθ(z)+

1

4σ8

∫∞0 gθ(z, r)(

zr − µ)4dr

gθ(z)

− 1

4σ8[

∫∞0 gθ(z, r)(

zr − µ)2dr

gθ(z)]2 +

1

2σ4. (30)

Finally, we compute ∂2

∂µ∂σ2 [ln gθ(z)]. Recalling (28), we can write

∂2

∂µ∂σ2[ln gθ(z)] =

∂

∂µ

[1

2σ4

∫∞0 gθ(z, r)(

zr − µ)2dr

gθ(z)

]=

1

2σ4gθ(z)

[−2

∫ ∞0

gθ(z, r)(z

r− µ)dr +

1

σ2

∫ ∞0

gθ(z, r)(z

r− µ)3dr

]− 1

2σ6

[∫∞0 gθ(z, r)(

zr − µ)dr][

∫∞0 gθ(z, r)(

zr − µ)2dr

]g2θ(z)

= − 1

σ4

∫∞0 gθ(z, r)(

zr − µ)dr

gθ(z)+

1

2σ6

∫∞0 gθ(z, r)(

zr − µ)3dr

gθ(z)

− 1

2σ6

[∫∞

0 gθ(z, r)(zr − µ)dr][

∫∞0 gθ(z, r)(

zr − µ)2dr]

g2θ(z)

. (31)

B.3 Lognormal distribution

We assume Y ∼ fθ(y) ∼ 1yσe− (ln y−µ)2

2σ2 , 0 < y < ∞, and θ = (µ, σ2). Then the joint pdf g(z, r)of (Z,R), the marginal pdf g(z) of Z and the conditional pdf g(r|z) of R, given Z = z, are givenby

gθ(z, r) ∼1

zσe−

(ln z−ln r−µ)2

2σ2 h(r),

gθ(z) ∼1

zσ

∫ ∞0

e−ln z−ln r−µ)2

2σ2 h(r)dr, (32)

gθ(r|z) =e−

(ln z−ln r−µ)2

2σ2 h(r)∫∞0 e−

(ln z−ln r−µ)22σ2

h(r)dr, (33)

for 0 < r < ∞ and 0 < z < ∞, and hence the complete data likelihood and log-likelihood can beexpressed as

L(θ|zc) ∼1

σne− 1

2σ2

∑ni=1(ln

ziri−µ)2

, (34)

`(θ|uc) = −n lnσ − 1

2σ2

n∑i=1

(lnziri− µ)2 (35)

= −n lnσ − 1

2σ2

n∑i=1

(lnziri

)2 +µ

σ2

n∑i=1

lnziri− nµ2

2σ2.

42

The EM steps for computing the MLE of θ based on z are as follows.E-step. We compute

Q(θ|θ(t)) = Eθ(t) [`(θ|uc)|uobs]

= −n lnσ − 1

2σ2

n∑i=1

[ψ2(θ(t), zi) +µ

σ2ψ1(θ(t), zi)]−

nµ2

2σ2, (36)

where ψ1(θ(t), zi) = Eθ(t) [lnziri|zi] and ψ2(θ(t), zi) = Eθ(t) [(ln

ziri

)2|zi], and these two quantities canbe readily computed based on the conditional pdf of R, given z, mentioned in (33).M-step. We note that

∂Q(θ|θ(t))

∂µ=

1

σ2

n∑i=1

ψ1(θ(t), zi)−nµ

σ2= 0

yields µ = 1n

∑ni=1 ψ1(θ(t), zi), and,

∂Q(θ|θ(t))

∂σ2= − n

2σ2+

1

2σ4

n∑i=1

ψ2(θ(t), zi)−µ

σ4

n∑i=1

ψ1(θ(t), zi) +nµ2

2σ4= 0

yields σ2 = 1n

∑ni=1 ψ2(θ(t), zi)− µ2. Thus the EM iterations are defined by

µ(t+1) =1

n

n∑i=1

ψ1(θ(t), zi), (σ(t+1))2 =1

n

n∑i=1

ψ2(θ(t), zi)− (µ(t+1))2. (37)

The observed Fisher information is computed by adding component-wise terms based on z1, ...., zn,with a generic term as follows. Since the marginal pdf of Z is given by (32), we obtain,

∂ ln gθ(z)

∂µ=

∫∞0 e−

(ln z−ln r−µ)2

2σ2ln z−ln r−µ

σ2 h(r)dr∫∞0 e−

(ln z−ln r−µ)22σ2 h(r)dr

, (38)

∂2 ln gθ(z)

∂µ2= − 1

σ2+

∫∞0 e−

(ln z−ln r−µ)2

2σ2 ( ln z−ln r−µσ2 )2h(r)dr∫∞

0 e−(ln z−ln r−µ)2

2σ2 h(r)dr

−

∫∞0 e−(ln z−ln r−µ)2

2σ2ln z−ln r−µ

σ2 h(r)dr∫∞0 e−


2

, (39)

43

and

∂2 ln gθ(z)

∂µ∂σ2= − 1

σ4×∫∞


2σ2 (ln z − ln r − µ)h(r)dr∫∞0 e−


+1

2σ6×∫∞


2σ2 (ln z − ln r − µ)3h(r)dr∫∞0 e−


− 1

2σ6

[∫∞0 e−

(ln z−ln r−µ)2

2σ2 (ln z − ln r − µ)h(r)dr ×∫∞


2σ2 (ln(z/r)− µ)2h(r)dr

][∫∞

0 e−(ln(z/r)−µ)2

2σ2 h(r)dr

]2 .

(40)

Finally, we get

∂ ln gθ(z)

∂σ2= − 1

2σ2+

1

2σ4

∫∞0 e−

(ln z−ln r−µ)2

2σ2 (ln z − ln r − µ)2h(r)dr∫∞0 e−


,

∂2 ln gθ(z)

(∂σ2)2=

1

2σ4+

1

σ6

∫∞0 e−

(ln z−ln r−µ)2

2σ2 (ln z − ln r − µ)2h(r)dr∫∞0 e−


+1

4σ8

∫∞0 e−

(ln z−ln r−µ)2

2σ2 (ln z − ln r − µ)4h(r)dr∫∞0 e−


− 1

4σ8

∫∞0 e−(ln z−ln r−µ)2

2σ2 (ln z − ln r − µ)2h(r)dr∫∞0 e−


2

. (41)

A customized or conjugate noise distribution is obtained as follows. Since Y follows a lognormaldistribution, implying lnY ∼ N(µ, σ2) and since Z = Y × R, we choose R ∼ lognormal with

lnR ∼ N(−ψ2

2 , ψ2). Then E(R) = 1 and var(R) = σ2

r = eψ2 − 1. This readily yields

lnZ ∼ N(µ− ψ2

2= µz, σ

2 + ψ2 = σ2z) (42)

Since µz = 1n

∑ni=1 ln zi and σ2

z = 1n

∑ni=1(ln zi − µz)2, we obtain µ = µz + ψ2

2 and σ2 = σ2z − ψ2.

Since the estimated variance-covariance matrix of (µz, σ2z) readily follows from the observed Fisher

information matrix for (µz, σ2z) given by

Iobs(µz, σ2z) =

(nσ2z

0

0 n2σ4z

). (43)

The estimated variances of meaningful and useful functions g(., .) of µz and σ2z can be easily obtained

by computing the required partial derivatives. We provide below examples of such functions.

44

1. g(µz, σ2z) = eµ+σ2

2 = eµz+ψ2

2 .

2. g(µz, σ2z) = e2µ+2σ2 − e2µ+σ2

= e2µz+2σ2z−ψ2 − e2µz+σ2

z .

3. g(µz, σ2z) = eµ+cσ = eµz+ψ2

2+c(σ2

z−ψ2)12 .

Note that the first and second functions are, respectively, the mean and variance of the originallognormal random variable Y , expressed in terms of the mean and variance of the random variablelnZ, specified in (42). Similarly, the third function is a percentile of Y . Recall once again thatdata are not available on Y , but observations are available on the perturbed quantity Z. In otherwords, inference on the above functions related to the distribution of Y can now be carried outusing the data on Z.

45

APPENDIX C: MLEs AND FISHER INFORMATION UNDER NOISE MULTIPLICATION OFEXTREME VALUES

Here we present details concerning the computation of the MLEs, and derivation of the Fisherinformation matrices in the context of the exponential, normal and lognormal distributions. Theseare required for the data analysis presented in Section 3 of the paper, dealing with noise multi-plication of the extreme values in the sample. In what follows, the notations used are as given inSection 3 of the paper.

C1. Exponential distributionWe consider the case of Y ∼ fθ(y) = 1

θe−y/θ, 0 < y <∞. Then the complete data log-likelihood

function can be expressed as

`(θ|y) = −n ln θ − 1

θ

n∑i=1

[∆ixi + (1−∆i)xiri

]. (44)

Case (i). We now describe the EM algorithm for computing the MLE of θ based on the data vi,obs.

E-step.


= −n ln θ − 1

θ

n∑i=1

{∆ixi + (1−∆i)xiEθ(t)

[r−1i |xi,∆i = 0

]}where

Eθ(t)[r−1i |xi,∆i = 0

]=

∫ xi/C0 exp

(− 1θ(t)

xir

)h(r)r−2dr∫ xi/C

0 exp(− 1θ(t)

xiω

)h(ω)ω−1dω

.

M-step. The value of θ that maximizes Q(θ|θ(t)) is readily computed, thus the following equationdefines the sequence of EM iterations:

θ(t+1) =1

n

n∑i=1

{∆ixi + (1−∆i)xiEθ(t)

[r−1i |xi,∆i = 0

]}(45)

with the expression for Eθ(t) [r−1i |xi,∆i = 0] as given above.

From Result 2 in Appendix A of the paper, the loglikelihood is

`(θ|x1, . . . , xn,∆1, . . . ,∆n) = −n ln θ − 1

θ

n∑i=1

xi∆i +

n∑i=1

(1−∆i) ln[

∫ x/C

0exp(−xi

rθ)h(r)r−1dr],

and hence the observed Fisher information is readily obtained based on the following:

∂2l(θ|x1, . . . , xn,∆1, . . . ,∆n)

∂θ2=

n

θ2− 2

θ3

n∑i=1

xi∆i −2

θ3

n∑i=1

(1−∆i)xi

∫ xi/C0 e−

xirθh(r)r2dr∫ xi/C

0 e−xirθh(r)r dr

+1

θ4

n∑i=1

(1−∆i)x2i

∫ xi/C0 e−


0 e−xirθh(r)r dr

− 1

θ4

n∑i=1

(1−∆i)x2i [

∫ xi/C0 e−


0 e−xirθh(r)r dr

]2. (46)

46

Case (ii). We now describe the EM algorithm for computing the MLE of θ based on the datavii,obs.E-step.


= −n ln θ − 1

θ

n∑i=1

{xiEθ(t) [∆i|xi] + xiEθ(t) [(1−∆i)r

−1i |xi]

},

where

Eθ(t) [∆i|xi] =I(xi < C) exp

[−xi/θ(t)

]I(xi < C) exp

[−xi/θ(t)

]+∫ xi/C

0 exp[− 1θ(t)

xiω

]h(ω)ω−1dω

,

and

Eθ(t) [(1−∆i)r−1i |xi] =

∫ xi/C0 exp

[− 1θ(t)

xiri

]h(ri)r

−2i dri

I(xi < C) exp[−xi/θ(t)

]+∫ xi/C

0 exp[− 1θ(t)

xiω

]h(ω)ω−1dω

.

M-step. Maximization of Q(θ|θ(t)) with respect to θ is readily accomplished, and hence we obtainthe following equation which defines the sequence of EM iterations:

θ(t+1) =1

n

n∑i=1

{xiEθ(t) [∆i|xi] + xiEθ(t) [(1−∆i)r

−1i |xi]

}, (47)

where the expressions for Eθ(t) [∆i|xi] and Eθ(t) [(1−∆i)r−1i |xi] are given above.

Recalling Result 4 given in Appendix A of the paper, the observed Fisher information in thiscase is based on the loglikelihood

`(θ|x1, . . . , xn) = −n ln θ +n∑i=1

ln[e−xiθ I(xi < C) +

∫ xiC

0e−

xirθh(r)

rdr] (48)

This gives

∂l(θ|x1, . . . , xn)

∂θ= −n

θ+

1

θ2

n∑i=1

xi[e−

xiθ I(xi < C) +

∫ xiC

0 e−xirθh(r)r2dr

kθ(xi)]. (49)

Hence we get

∂2l(θ|x1, . . . , xn)

∂θ2=

n

θ2− 2

θ3

n∑i=1

xi[e−

xiθ I(xi < C) +

∫ xiC

0 e−xirθh(r)r2dr

kθ(xi)]

+1

θ4

n∑i=1

x2i [e−

xiθ I(xi < C) +

∫ xiC

0 e−xirθh(r)r3dr

kθ(xi)]

− 1

θ4

n∑i=1

x2i [e−

xiθ I(xi < C) +

∫ xiC

0 e−xirθh(r)r2dr

kθ(xi)]2. (50)

47

C2. Normal distributionWe next consider the case Y ∼ fθ(y) = 1√

2πσ2exp

[− (y−µ)2

2σ2

], −∞ < y <∞, where θ = (µ, σ2).

Then the complete data likelihood function is

L(θ|vc) =n∏i=1

{1√

2πσ2exp

[−(xi − µ)2

2σ2

]}∆i{

1√2πσ2

exp

[−(xiri − µ)2

2σ2

]}1−∆i

∝n∏i=1

{1√σ2

exp

[−(xi − µ)2

2σ2

]}∆i{

1√σ2

exp

[−

(xiri − µ)2

2σ2

]}1−∆i

= (σ2)−n/2n∏i=1

{exp

[−(xi − µ)2

2σ2

]}∆i{

exp

[−

(xiri − µ)2

2σ2

]}1−∆i

,

and hence we get the complete data log-likelihood function as

`(θ|vc) = −n2

ln(σ2)− 1

2σ2

n∑i=1

{∆i(xi − µ)2 + (1−∆i)

(xiri− µ

)2}

= −n2

ln(σ2)− nµ2

2σ2− 1

2σ2

n∑i=1

{∆i

[(xi)

2 − 2µxi]

+ (1−∆i)

[(xiri

)2 − 2µxiri

]}.

Case (i). The EM algorithm for computing the MLE of θ based on (x1,∆1), . . . , (xn,∆n) is asfollows.

E-step.

Q(θ|θ(t)) = Eθ(t) [`(θ|vc)|vi,obs] = −n2

ln(σ2)− nµ2

2σ2− 1

2σ2

n∑i=1

{∆i[(xi)

2 − 2µxi]}

− 1

2σ2

n∑i=1

{(1−∆i)(Eθ(t) [(xiri

)2|xi,∆i = 0]− 2µEθ(t) [(xiri

)|xi,∆i = 0])}

where

Eθ(t)

[(xiri

)2|xi,∆i = 0

]= x2

i

∫ xi/C0 exp

[− (xi/r)−µ(t)}2

2(σ(t))2

]h(r)r3dr∫ xi/C

0 exp[− (xi/ω)−µ(t)}2

2(σ(t))2

]h(ω)ω dω

≡ x2iψ1(θ(t), xi),

Eθ(t)

[(xiri

)|xi,∆i = 0

]= xi

∫ xi/C0 exp

[− (xi/r)−µ(t)}2

2(σ(t))2

]h(r)r2dr∫ xi/C

0 exp[− (xi/ω)−µ(t)}2

2(σ(t))2

]h(ω)ω dω

≡ xiψ2(θ(t), xi).

M-step. By maximizing Q(θ|θ(t)) with respect to θ, we obtain the following equations which definethe sequence of EM iterations:

µ(t+1) =1

n

n∑i=1

{∆ixi + (1−∆i)xiψ2(θ(t), xi)

},

(σ(t+1))2 =1

n

(n∑i=1

{∆i(xi)

2 + (1−∆i)x2iψ1(θ(t), xi)

})− (µ(t+1))2,

48

where the expressions for ψ1(θ(t), xi) and ψ2(θ(t), xi) are given above. When ∆i = 1, ψ1(θ(t), xi)and ψ2(θ(t), xi) are defined as equal to 0.

To compute the observed Fisher information, note that the loglikelihood based on one observa-tion x can be expressed as

`(θ|x,∆) = lnL(θ|x,∆) ∼ −1

2ln(σ2)− 1

2σ2∆(x−µ)2 + (1−∆) ln

[∫ xC

0e−

12σ2

(xr−µ)2 h(r)

rdr

]. (51)

This yields

∂`(θ|x,∆)

∂µ=

∆(x− µ)

σ2+

(1−∆)

σ2

[∫ xC

0 e−1

2σ2(xr−µ)2(xr − µ)h(r)

r dr∫ xC

0 e−1

2σ2(xr−µ)2 h(r)

r dr

]

=∆(x− µ)

σ2− µ(1−∆)

σ2+

(1−∆)

σ2

[∫ xC

0 e−1

2σ2(xr−µ)2(xr )h(r)

r dr∫ xC

0 e−1

2σ2(xr−µ)2 h(r)

r dr

]. (52)

Hence we get

∂2`(θ|x,∆)

∂µ2= − 1

σ2+

(1−∆)

σ4

[∫ xC

0 e−1

2σ2(xr−µ)2(xr − µ)(xr )h(r)

r dr∫ xC

0 e−1

2σ2(xr−µ)2 h(r)

r dr

−(∫ xC

0 e−1

2σ2(xr−µ)2(xr − µ)h(r)

r dr)(∫ xC

0 e−1


r dr)

(∫ xC

0 e−1

2σ2(xr−µ)2 h(r)

r dr)2

]. (53)

From (52), we get

∂2`(θ|x,∆)

∂µ∂σ2= −x∆− µ

σ4− (1−∆)

σ4[

∫ xC

0 e−1


r dr∫ xC

0 e−1

2σ2(xr−µ)2 h(r)

r dr]

+(1−∆)

2σ6

[∫ xC

0 e−1

2σ2(xr−µ)2(xr − µ)2(xr )h(r)

r dr∫ xC

0 e−1

2σ2(xr−µ)2 h(r)

r dr

−(∫ xC

0 e−1

2σ2(xr−µ)2(xr − µ)2 h(r)

r dr)(∫ xC

0 e−1


r dr)

(∫ xC

0 e−1

2σ2(xr−µ)2 h(r)

r dr)2

]. (54)

Lastly, we note that

∂`(θ|x,∆)

∂σ2= − 1

2σ2+

∆(x− µ)2

2σ4+

(1−∆)

2σ4

[∫ xC

0 e−1

2σ2(xr−µ)2(xr − µ)2 h(r)

r dr∫ xC

0 e−1

2σ2(xr−µ)2 h(r)

r dr

]. (55)

Hence we get

∂2`(θ|x,∆)

∂(σ2)2=

1

2σ4− ∆(x− µ)2

σ6− (1−∆)

σ6

[∫ xC

0 e−1

2σ2(xr−µ)2(xr − µ)2 h(r)

r dr∫ xC

0 e−1

2σ2(xr−µ)2 h(r)

r dr

]

+(1−∆)

4σ8

[∫ xC

0 e−1

2σ2(xr−µ)2(xr − µ)4 h(r)

r dr∫ xC

0 e−1

2σ2(xr−µ)2 h(r)

r dr− (

∫ xC

0 e−1

2σ2(xr−µ)2(xr − µ)2 h(r)

r dr∫ xC

0 e−1

2σ2(xr−µ)2 h(r)

r dr)2

]. (56)

49

Case (ii). The EM algorithm for computing the MLE of θ based on (x1, . . . , xn) is as follows.

E-step.


= −n2

ln(σ2)− nµ2

2σ2− 1

2σ2

n∑i=1

{(xi)

2 − 2µxi}Eθ(t) [∆i|xi]}

− 1

2σ2

n∑i=1

{Eθ(t) [(1−∆i)(

xiri

)2|xi]− 2µEθ(t) [(1−∆i)(xiri

)|xi]}

where


[− (xi−µ(t))2

2(σ(t))2

]I(xi < C) exp

[− (xi−µ(t))2

2(σ(t))2

]+∫ xi/C

0 exp[−{(xi/ω)−µ(t)}2

2(σ(t))2

]h(ω)ω dω

≡ ψ1(θ(t), xi).

Eθ(t) [(1−∆i)(xiri

)2|xi] = x2i

∫ xi/C0 exp

[−{ln(xi/r)−µ(t)}2

2(σ(t))2

]h(r)r3dr

I(xi < C) exp[−{xi−µ

(t)}22(σ(t))2

]+∫ xi/C

0 exp[−{(xi/ω)−µ(t)}2

2(σ(t))2

]h(ω)ω dω

≡ x2iψ2(θ(t), xi).

Eθ(t) [(1−∆i)(xiri

)|xi] = xi

∫ xi/C0 (xir ) exp

[−{(xi/r)−µ

(t)}22(σ(t))2

]h(r)r2dr

I(xi < C) exp[−{xi−µ

(t)}22(σ(t))2

]+∫ xi/C

0 exp[−{(xi/ω)−µ(t)}2

2(σ(t))2

]h(ω)ω dω

≡ xiψ3(θ(t), xi).

Hence we can write

Q(θ|θ(t)) = −n2

ln(σ2)− nµ2

2σ2− 1

2σ2

n∑i=1

{(xi)

2 − 2µxi}ψ1(θ(t), xi)

− 1

2σ2

n∑i=1

(x2iψ2(θ(t), xi)− 2µxiψ3(θ(t), xi).


µ(t+1) =1

n

n∑i=1

[xi(ψ1(θ(t), xi) + ψ3(θ(t), xi))

],

(σ(t+1))2 =1

n

n∑i=1

[x2i (ψ1(θ(t), xi) + ψ2(θ(t), xi))

]− (µ(t+1))2

50

where the expressions for ψ1(θ(t), xi), ψ1(θ(t), xi) and ψ1(θ(t), xi) are given above.To compute the observed Fisher information in this case, we can express the pdf of a single

observation x as

kθ(x) =1

σφ(x− µσ

)I(x < C) +

∫ xC

0φ

( xr − µσ

)I(x > 0)

h(r)

rdr, (57)

where φ(.) is the standard normal pdf . This yields

∂kθ(x)

∂µ=x− µσ3

φ

(x− µσ

)I(x < C) +

∫ xC

0

xr − µσ3

φ

( xr − µσ

)I(x > 0)

h(r)

rdr, (58)

and hence

∂2kθ(x)

∂µ2= −kθ(x)

σ2+

(x− µ)2

σ4

1

σφ

(x− µσ

)I(x < C)

+

∫ xC

0

(xr − µ)2

σ4

1

σφ

( xr − µσ

)I(x > 0)

h(r)

rdr. (59)

We also have

∂2kθ(x)

∂µ∂σ2= −3(x− µ)

2σ5φ

(x− µσ

)I(x < C)

− 3

2σ5

∫ xC

0(x

r− µ)φ

( xr − µσ

)I(x > 0)

h(r)

rdr

+x− µσ3

[− 1

2σ2+

(x− µ)2

2σ4]φ

(x− µσ

)I(x < C)

+

∫ xC

0

xr − µσ3

[− 1

2σ2+

(xr − µ)2

2σ4

]φ

( xr − µσ

)I(x > 0)

h(r)

rdr. (60)

Lastly, we can write

∂kθ(x)

∂σ2= − 1

2σ2kθ(x) +

1

σφ

(x− µσ

)[− 1

2σ2+

(x− µ)2

2σ4

]I(x < C)

+

∫ xC

0φ

( xr − µσ

)I(x > 0)

[− 1

2σ2+

(xr − µ)2

2σ4

]h(r)

rdr, (61)

so that

∂2kθ(x)

∂(σ2)2=

1

2σ4kθ(x)− 1

2σ2

∂kθ(x)

∂σ2+ φ

(x− µσ

)[3

4σ5− 5(x− µ)2

4σ7

]I(x < C)

+

∫ xC

0φ

( xr − µσ

)I(x > 0)

[3

4σ5−

5(xr − µ)2

4σ7

]h(r)

rdr

+1

σφ

(x− µσ

)[− 1

2σ2+

(x− µ)2

2σ4

]2

I(x < C)

+

∫ xC

0φ

( xr − µσ

)I(x > 0)

[− 1

2σ2+

(xr − µ)2

2σ4

]2h(r)

rdr. (62)

51

We can use the above expressions to compute the elements of the observed Fisher informationmatrix.

C3. Lognormal distribution

We now assume that Y ∼ fθ(y) = 1

y√

2πσ2exp

[− (ln y−µ)2

2σ2

], 0 < y <∞, where θ = (µ, σ2). Then

the complete data likelihood function is

L(θ|vc) =

n∏i=1

{1

xi√

2πσ2exp

[−(lnxi − µ)2

2σ2

]}∆i{

ri

xi√

2πσ2exp

[−(ln(xi/ri)− µ)2

2σ2

]}1−∆i

∝n∏i=1

{1√σ2

exp

[−(lnxi − µ)2

2σ2

]}∆i{

1√σ2

exp


2σ2

]}1−∆i

= (σ2)−n/2n∏i=1

{exp

[−(lnxi − µ)2

2σ2

]}∆i{

exp


2σ2

]}1−∆i

,

and we get the complete data log-likelihood function as

`(θ|vc) = −n2

ln(σ2)− 1

2σ2

n∑i=1

{∆i(lnxi − µ)2 + (1−∆i)(ln

xiri− µ)2

}

= −n2

ln(σ2)− nµ2

2σ2− 1

2σ2

n∑i=1

{∆i[(lnxi)

2 − 2µ lnxi] + (1−∆i)[(lnxiri

)2 − 2µ lnxiri

]

}.

Case (i). The EM algorithm for computing the MLE of θ based on (x1,∆1), . . . , (xn,∆n) is asfollows.

E-step.


= −n2

ln(σ2)− nµ2

2σ2− 1

2σ2

n∑i=1

{∆i[(lnxi)

2 − 2µ lnxi]}

− 1

2σ2

n∑i=1

{(1−∆i)

(Eθ(t) [(ln

xiri

)2|xi,∆i = 0]− 2µEθ(t) [(lnxiri

)|xi,∆i = 0]

)}where

Eθ(t) [(lnxiri

)2|xi,∆i = 0] =

∫ xi/C0 (ln xi

r )2 exp[−{ln(xi/r)−µ(t)}2

2(σ(t))2

]h(r)dr∫ xi/C

0 exp[−{ln(xi/ω)−µ(t)}2

2(σ(t))2

]h(ω)dω

,

Eθ(t) [(lnxiri

)|xi,∆i = 0] =

∫ xi/C0 (ln xi

r ) exp[−{ln(xi/r)−µ(t)}2

2(σ(t))2

]h(r)dr∫ xi/C


2(σ(t))2

]h(ω)dω

.

M-step. By maximizing Q(θ|θ(t)) with respect to θ, we obtain the following equations which define

52

the sequence of EM iterations:

µ(t+1) =1

n

n∑i=1

{∆i lnxi + (1−∆i)Eθ(t) [(ln

xiri

)|xi,∆i = 0]

},

(σ(t+1))2 =1

n

(n∑i=1

{∆i(lnxi)

2 + (1−∆i)Eθ(t) [(lnxiri

)2|xi,∆i = 0]

}− n(µ(t+1))2

),

where the expressions for Eθ(t) [(lnxiri

)2|xi,∆i = 0] and Eθ(t) [(lnxiri

)|xi,∆i = 0] are given above.To compute the observed Fisher information in this case, recalling Result 2 given in Appendix

A of the paper, a direct computation using the above formulas shows that for a generic term withx and ∆,

∂l(θ)

∂µ=

(lnx− µ)∆

σ2+ (1−∆)

∫ xC

0 e−(ln x−ln r−µ)2

2σ2(lnx−ln r−µ)2

σ2 h(r)dr∫ xC


2σ2 h(r)dr

(63)

leading to

∂2l(θ)

∂µ2= − 1

σ2+ (1−∆)[

∫ xC


2σ2 ( (lnx−ln r−µ)2

σ2 )2h(r)dr∫ xC


2σ2 h(r)dr

−{∫ xC


2σ2(lnx−ln r−µ)2

σ2 h(r)dr∫ xC


2σ2 h(r)dr

}2]. (64)

Likewise,

∂2l(θ)

∂µ∂σ2= −(lnx− µ)∆

σ4+ (1−∆)[−

∫ xC


2σ2 ( (lnx−ln r−µ)2

σ4 )h(r)dr∫ xC


2σ2 h(r)dr

+1

2σ6

∫ xC


2σ2 (lnx− ln r − µ)3h(r)dr∫ xC


2σ2 h(r)dr

− 1

2σ6

∫ xC


2σ2 (lnx− ln r − µ)h(r)dr∫ xC




2σ2 h(r)dr2

].(65)

53

Finally,

∂2l(θ)

(∂σ2)2=

1

2σ4− (lnx− µ)2∆

σ6+ (1−∆)[− 1

σ6

∫ xC




2σ2 h(r)dr

+1

4σ8

1

2σ6

∫ xC




2σ2 h(r)dr

− 1

4σ8{∫ xC




2σ2 h(r)dr

}2]. (66)

Case (ii). The EM algorithm for computing the MLE of θ based on (x1, . . . , xn) is as follows.

E-step.


= −n2

ln(σ2)− nµ2

2σ2− 1

2σ2

n∑i=1

{{(lnxi)2 − 2µ lnxi}Eθ(t) [∆i|xi]

}− 1

2σ2

n∑i=1

{Eθ(t) [(1−∆i)(ln

xiri

)2|xi]− 2µEθ(t) [(1−∆i)(lnxiri

)|xi]}

where


[− (lnxi−µ(t))2

2(σ(t))2

]I(xi < C) exp

[− (lnxi−µ(t))2

2(σ(t))2

]+∫ xi/C

0 exp[− (ln(xi/ω)−µ(t))2

2(σ(t))2

]h(ω)dω

.

Eθ(t) [(1−∆i)(lnxiri

)2|xi]

=

∫ xi/C0 (ln xi

r )2 exp[−{ln(xi/r)−µ(t)}2

2(σ(t))2

]h(r)dr

I(xi < C) exp[−{lnxi−µ

(t)}22(σ(t))2

]+∫ xi/C


2(σ(t))2

]h(ω)dω

,

Eθ(t) [(1−∆i)(lnxiri

)|xi] =

∫ xi/C0 (ln xi

r ) exp[−{ln(xi/r)−µ(t)}2

2(σ(t))2

]h(r)dr

I(xi < C) exp[−{lnxi−µ

(t)}22(σ(t))2

]+∫ xi/C


2(σ(t))2

]h(ω)dω

.


µ(t+1) =1

n

n∑i=1

{(lnxi)Eθ(t) [∆i|xi] + Eθ(t) [(1−∆i)(ln

xiri

)|xi]},

(σ(t+1))2 =1

n

(n∑i=1

{(lnxi)

2Eθ(t) [∆i|xi] + Eθ(t) [(1−∆i)(lnxiri

)2|xi]}− n(µ(t+1))2

),

54

where formulas for Eθ(t) [∆i|xi], Eθ(t) [(1 − ∆i)(lnxiri

)2|xi], and Eθ(t) [(1 − ∆i)(lnxiri

)|xi] are givenabove.

To compute the observed Fisher information in this case, note that (by Result 3 of AppendixA of the paper) the loglikelihood based on a generic x is given by ln k(x, θ) where

k(x, θ) ∼ 1

σ[e−

(ln x−µ)2

2σ2 I(x < C) +

∫ xC

0e−

(ln x−ln r−µ)2

2σ2 I(x > 0)h(r)dr]. (67)

Hence we get

∂ ln k(x, θ)

∂µ=e−

(ln x−µ)2

2σ2lnx−µσ2 I(x < C) +

∫ xC


2σ2 I(x > 0) lnx−ln r−µσ2 h(r)dr

e−(ln x−µ)2

2σ2 I(x < C) +∫ xC


2σ2 I(x > 0)h(r)dr

= − µ

σ2+

1

σ2

e− (ln x−µ)2

2σ2 (lnx)I(x < C) +∫ xC


2σ2 I(x > 0)(lnx− ln r)h(r)dr

e−(ln x−µ)2

2σ2 I(x < C) +∫ xC


2σ2 I(x > 0)h(r)dr

(68)

This readily yields

∂2 ln k(x, θ)

∂µ2= − 1

σ2

+1

σ4

e− (ln x−µ)2

2σ2 (lnx)(lnx− µ)I(x < C) +∫ xC

0e−

(ln(x/r)−µ)2

2σ2 I(x > 0) ln(x/r)(ln(x/r)− µ)h(r)dr

e−(ln x−µ)2

2σ2 I(x < C) +∫ xC

0e−

(ln x−ln r−µ)22σ2 I(x > 0)h(r)dr

− 1

σ4

{[e−

(ln x−µ)2

2σ2 (lnx)I(x < C) +

∫ xC

0

e−(ln(x/r)−µ)2

2σ2 I(x > 0) ln(x/r)h(r)dr

]

×

[e−

(ln x−µ)2

2σ2 (lnx− µ)I(x < C) +

∫ xC

0

e−(ln(x/r)−µ)2

2σ2 I(x > 0)(ln(x/r)− µ)h(r)dr

]

×

[e−

(ln x−µ)2

2σ2 I(x < C) +

∫ xC

0

e−(ln(x/r)−µ)2

2σ2 I(x > 0)h(r)dr

]−2 . (69)

From (68), we get

∂2 ln k(x, θ)

∂µ∂σ2=

µ

σ4− 1

σ4

e− (ln x−µ)2

2σ2 (lnx)I(x < C) +∫ xC

0e−

(ln(x/r)−µ)2


e−(ln x−µ)2

2σ2 I(x < C) +∫ xC

0e−

(ln(x/r)−µ)22σ2 I(x > 0)h(r)dr

+

1

2σ6

e− (ln x−µ)2

2σ2 (lnx)(lnx− µ)2I(x < C) +∫ xC

0e−

(ln(x/r)−µ)2

2σ2 I(x > 0) ln(x/r)(ln(x/r)− µ)2h(r)dr

e−(ln x−µ)2

2σ2 I(x < C) +∫ xC

0e−

(ln(x/r)−µ)22σ2 I(x > 0)h(r)dr

− 1

2σ6

{[e−

(ln x−µ)2

2σ2 (lnx)I(x < C) +

∫ xC

0

e−(ln(x/r)−µ)2


]

×

[e−

(ln x−µ)2

2σ2 (lnx− µ)2I(x < C) +

∫ xC

0

e−(ln(x/r)−µ)2

2σ2 I(x > 0)(ln(x/r)− µ)2h(r)dr

]

×

[e−

(ln x−µ)2

2σ2 I(x < C) +

∫ xC

0

e−(ln(x/r)−µ)2

2σ2 I(x > 0)h(r)dr

]−2 . (70)

55

On the other hand, we get

∂ ln k(x, θ)

∂σ2= − 1

2σ2

+1

2σ4

e− (ln x−µ)2

2σ2 (lnx− µ)2I(x < C) +∫ xC


2σ2 I(x > 0)(lnx− ln r − µ)2h(r)dr

e−(ln x−µ)2

2σ2 I(x < C) +∫ xC


2σ2 I(x > 0)h(r)dr

,and hence,

∂2 ln k(x, θ)

∂(σ2)2=

1

2σ4

− 1

σ6

e− (ln x−µ)2

2σ2 (lnx− µ)2I(x < C) +∫ xC



e−(ln x−µ)2

2σ2 I(x < C) +∫ xC


2σ2 I(x > 0)h(r)dr

+

1

4σ8

e− (ln x−µ)2

2σ2 (lnx− µ)4I(x < C) +∫ xC



e−(ln x−µ)2

2σ2 I(x < C) +∫ xC


2σ2 I(x > 0)h(r)dr

− 1

4σ8

e− (ln x−µ)2

2σ2 (lnx− µ)2I(x < C) +∫ xC



e−(ln x−µ)2

2σ2 I(x < C) +∫ xC


2σ2 I(x > 0)h(r)dr

2

.

56

APPENDIX D: MLEs AND FISHER INFORMATION UNDER TOP CODING

In this Appendix, we present the MLEs, along with the expressions for the observed Fisherinformation matrix for the three parametric models: exponential, normal and lognormal.

D1. Exponential distributionAssume y1, . . . , yn is a random sample from the exponential distribution fθ(y) = 1

θe−y/θ, y > 0.

Let xi = min{yi, C}, and ∆i = I(yi ≤ C), for i = 1, . . . , n. The likelihood for θ based on theobserved data wobs = (x1, . . . , xn,∆1, . . . ,∆n) is given by

L(θ|wobs) = (1

θ)∑ni=1 ∆i × e−

∑ni=1 xiθ .

It is easily seen that the MLE of θ is

θMLE =

∑ni=1 xi∑ni=1 ∆i

,

and the observed Fisher information is

(∑n

i=1 ∆i)2∑n

i=1 xi.

D2. Normal distributionLet C denote the known top code threshold. Assume y1, . . . , yn ∼ iid N(µ, σ2) and let xi =

min{yi, C} and ∆i = I(yi ≤ C), i = 1, . . . , n. In this case we can denote the complete and observeddata, respectively, as

wc = (y1, . . . , yn), wobs = (x1, . . . , xn,∆1, . . . ,∆n).

Obviously the complete data loglikelihood is given by

`(µ, σ2|wc) ∼ −n

2ln(σ2)− 1

2σ2

n∑i=1

y2i +

µ

σ2

n∑i=1

yi −nµ2

2σ2. (71)

The MLE of (µ, σ2) based on the observed data wobs can be computed via the EM algorithm. TheE and M steps are as follows.E-step. We have

Q(θ|θ(t)) = Eθ(t) [`(µ, σ2|wc)|wobs]

= −n2

ln(σ2)− 1

2σ2

n∑i=1

Eθ(t) [y2i |wobs] +

µ

σ2

n∑i=1

Eθ(t) [yi|wobs]−nµ2

2σ2

= −n2

ln(σ2)− 1

2σ2

n∑i=1

Eθ(t) [y2i |xi,∆i] +

µ

σ2

n∑i=1

Eθ(t) [yi|xi,∆i]−nµ2

2σ2. (72)

57

where

Eθ(t) [yi|xi,∆i] = xi∆i + (1−∆i)

∫∞C

yσ(t)φ

(y−µ(t)σ(t)

)dy

1− Φ(C−µ(t)σ(t)

) , (73)

Eθ(t) [y2i |xi,∆i] = x2

i∆i + (1−∆i)

∫∞C

y2

σ(t)φ(y−µ(t)σ(t)

)dy

1− Φ(C−µ(t)σ(t)

) . (74)

It is easy to verify by direct computation that

ψ1(θ) ≡∫∞C

yσφ(y−µ

σ

)dy

1− Φ(C−µσ

) = µ+ σφ(C−µσ

)1− Φ

(C−µσ

) (75)

ψ2(θ) ≡∫∞C

y2

σ φ(y−µ

σ

)dy

1− Φ(C−µσ

) = µ2 + σ2 + σ(µ+ C)φ(C−µσ

)1− Φ

(C−µσ

) . (76)

We can therefore simplify Q(θ|θ(t)) as

Q(θ|θ(t)) = −n2

ln(σ2)− 1

2σ2

n∑i=1

x2i∆i −

1

2σ2ψ2(θ(t))

n∑i=1

(1−∆i) (77)

+µ

σ2

n∑i=1

xi∆i +µ

σ2ψ1(θ(t))

n∑i=1

(1−∆i)−nµ2

2σ2.

M-step. Using (77) we can simplify ∂Q(θ|θ(t)∂µ = 0 and ∂Q(θ|θ(t))

∂σ2 = 0, resulting in the solution

µ =1

n[

n∑i=1

xi∆i + ψ1(θ(t))

n∑i=1

(1−∆i)],

σ2 =1

n[

n∑i=1

x2i∆i + ψ2(θ)

n∑i=1

(1−∆i)]− µ2. (78)

Thus the EM algorithm in this case is defined by the following iterations:

µ(t+1) =1

n[

n∑i=1

xi∆i + ψ1(θ(t))n∑i=1

(1−∆i)],

(σ(t+1))2 =1

n[n∑i=1

x2i∆i + ψ2(θ(t))

n∑i=1

(1−∆i)]− [µ(t+1)]2. (79)

We now discuss the computation of observed Fisher information. Since the likelihood functionfor θ based on wobs is given by

L(θ|wobs) ∼n∏i=1

([1

σφ

(xi − µσ

)]∆i[Φ

(xi − µσ

)]1−∆i),

58

the loglikelihood `(θ) is obtained as

`(θ|wobs) ∼ − ln(σ)

(n∑i=1

∆i

)+

n∑i=1

∆i lnφ

(xi − µσ

)+

n∑i=1

(1−∆i) ln Φ

(xi − µσ

). (80)

In order to derive the elements of the Fisher information matrix, we shall use the following prop-erties: φ′(u) = −uφ(u) and

∂

∂σ2

[φ(xi−µ

σ

)Φ(xi−µσ )

]=

1

2σ2

φ(xi−µ

σ

) (xi−µσ

)Φ(xi−µ

σ

) [(xi − µσ

)−φ(xi−µ

σ

)Φ(xi−µ

σ

)] ,where φ(u) is the standard normal pdf , and Φ(u) = 1 − Φ(u), where Φ(u) is the standard normalcdf . By straightforward algebra, we get

∂`(θ|wobs)

∂µ= − 1

σ

n∑i=1

∆iφ′(xi−µ

σ

)φ(xi−µσ )

+1

σ

n∑i=1

(1−∆i)φ(xi−µ

σ

)Φ(xi−µ

σ

)=

1

σ2

n∑i=1

∆i(xi − µ) +1

σ

n∑i=1


σ

)Φ(xi−µ

σ

) ,∂`(θ|wobs)

∂σ2= − 1

2σ2

n∑i=1

∆i +1

2σ4

n∑i=1

∆i(xi − µ)2 +1

2σ3

n∑i=1

(1−∆i)(xi − µ)φ(xi−µ

σ

)Φ(xi−µ

σ

) , (81)

and hence,

∂2`(θ|wobs)

∂µ2= − 1

σ2

n∑i=1

∆i +1

σ2

n∑i=1

(1−∆i)

(xi − µσ

)φ(xi−µ

σ

)Φ(xi−µ

σ

) − 1

σ2

n∑i=1

(1−∆i)

[φ(xi−µ

σ

)Φ(xi−µ

σ

)]2

,

∂2`(θ|wobs)

∂µ∂σ2= − 1

σ4

n∑i=1

∆i(xi − µ)− 1

2σ3

n∑i=1


σ

)Φ(xi−µ

σ

) +1

σ

n∑i=1

(1−∆i)∂

σ2

[φ(xi−µ

σ

)Φ(xi−µ

σ

)] ,∂2`(θ|wobs)

∂(σ2)2=

1

2σ4

n∑i=1

∆i −1

σ6

n∑i=1

∆i(xi − µ)2 − 3

4σ5

n∑i=1

(1−∆i)(xi − µ)φ

(xi−µσ

)Φ(xi−µ

σ

)+

1

4σ4

n∑i=1

(1−∆i)u3i

[φ(xi−µ

σ

)Φ(xi−µ

σ

)]− 1

4σ4

n∑i=1

(1−∆i)u2i

[φ(xi−µ

σ

)Φ(xi−µ

σ

)]2

. (82)

D3. Lognormal distributionWe once again use the property that if y1, . . . , yn is a random sample from a lognormal distri-

bution, then ln y1, . . ., ln yn is a sample from a normal distribution. We can now apply the normaltheory results given in the previous section upon replacing yi by ln yi and C by lnC.

59

APPENDIX E: SIMULATION RESULTS

In this Appendix, we present extensive numerical results for comparing multiple imputationand noise multiplication for the exponential, normal and lognormal models. We consider the caseof fully masked samples (where all values are replaced with multiply imputed or noise multiplieddata), as well as partially masked samples (where the top coded data are replaced with multiplyimputed or noise multiplied data).

The notations used in the tables are as given in Section 6 of the paper. The notation CD denotesthe case of complete data without any masking. MI denotes multiple imputation used to generatefully masked data. MI.C and MI.D are used to denote parametric multiple imputation based on thecomplete data, and based on the deleted data, respectively. These are explained in Section 4.2. Inthe case of a top coded sample with a top coding threshold C, values beyond a threshold CI werealso replaced with imputed values, where CI < C, following An and Little (2007). Two choices weremade for CI following the recipe prescribed by An and Little (2007). If nS denotes the number ofvalues beyond the top coding threshold C, the two choices of CI correspond to thresholds for which2nS values in the sample are larger, and 4nS values are larger. Since these values approximatelycorrespond to the 90th and 80th percentiles of the distribution when C is the 95th percentile, theseare denoted (following An and Little, 2007) by MI.C90 and MI.C80, respectively, in the case ofmultiple imputation based on the complete data, and denoted by MI.D90 and MI.D80, respectively,in the case of multiple imputation based on the deleted data. Throughout we have used 50 sets ofimputed values under MI.

The notations used under noise multiplication are as follows. We have used either a Uniform(1−ε, 1 + ε) distribution or a customized prior distribution for the noise multiplication. In the case ofUniform(1 − ε, 1 + ε), the notation NM10U denotes noise multiplication with the above uniformdistribution under the choice ε = 0.10. NM20U, NM30U etc. are similarly defined. Furthermore,NM10C, NM20C etc. are the corresponding notations under the customized prior whose mean andvariance match those of the respective uniform distributions. Additional notation used under theuniform noise distribution Uniform(1 − ε, 1 + ε) is as follows. For the choice ε = 0.1, NM10.1 andNM10.2 represent the case of observing the (xi,∆i)s, or only the xis; these are Case (i) and Case(ii) explained in Section 3 for the three different distributions. NM20.1, NM20.2 etc. are similarlydefined. In addition to the above notations, entries in the tables corresponding to CENS denoteresults from the analysis of the censored data, i.e., the top coded data.

Table E1 − Table E36 are for comparing fully masked samples where all values have beenprotected, and are as follows:

Table E1 − Table E3: Exponential distribution

Table E4 − Table E21: Lognormal distribution

Table E22 − Table E36: Normal distribution

Table E37 − Table E84 are for comparing partially masked samples where the extreme valueshave been protected, and are as follows:

Table E37 − Table E40: Exponential distribution

Table E41 − Table E64: Lognormal distribution

Table E65 − Table E84: Normal distribution

60

Tab

leE

1:In

fere

nce

for

the

mea

nb

ased

onfu

lly

mas

ked

Exp

onen

tial

(mea

n=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

183.7

4-3

.17

183.

7318

1.99

93.1

81.

000

140.

09-0

.43

140.

1014

1.36

94.2

81.

000

MI

211.8

568

.36

200.

5420

5.74

96.0

81.

130

154.

3440

.56

148.

9315

3.12

96.1

01.

083

NM

10U

184.2

9-3

.23

184.

2818

2.57

93.1

41.

003

140.

65-0

.12

140.

6714

1.86

94.3

41.

004

NM

10C

184.1

4-2

.91

184.

1418

2.63

92.9

81.

003

140.

78-0

.24

140.

7914

1.85

94.2

01.

003

NM

20U

186.0

7-2

.85

186.

0618

4.36

93.0

21.

013

141.

580.

1414

1.60

143.

2593

.98

1.01

3N

M20C

186.2

4-2

.12

186.

2518

4.46

92.9

81.

014

141.

620.

0314

1.64

143.

2294

.18

1.01

3N

M30U

191.1

7-0

.31

191.

1918

7.62

92.9

61.

031

144.

860.

4614

4.88

145.

4794

.42

1.02

9N

M30C

188.9

6-0

.80

188.

9718

7.37

93.0

81.

030

144.

410.

6214

4.42

145.

4094

.14

1.02

9N

M40U

193.5

50.

4019

3.57

191.

5193

.02

1.05

214

7.19

1.28

147.

2014

8.49

94.1

01.

050

NU

40C

193.0

10.

2319

3.03

190.

9892

.84

1.04

914

7.03

0.40

147.

0414

8.06

94.3

81.

047

NM

50U

198.2

8-0

.96

198.

3019

5.90

92.7

21.

076

152.

380.

5915

2.40

151.

9794

.22

1.07

5N

M50C

199.1

11.

1819

9.13

195.

1092

.94

1.07

215

0.62

2.77

150.

6115

1.49

94.1

41.

072

NM

60U

202.4

0-0

.74

202.

4220

1.48

92.6

01.

107

155.

622.

5415

5.61

156.

5194

.40

1.10

7N

M60C

201.0

22.

9220

1.02

199.

7092

.94

1.09

715

5.03

4.03

154.

9915

5.00

93.9

41.

096

NM

70U

210.8

04.

6521

0.77

209.

0192

.80

1.14

816

1.33

4.94

161.

2716

1.82

94.6

01.

145

NM

70C

205.9

13.

2120

5.90

204.

0493

.18

1.12

115

7.70

3.59

157.

6815

8.30

93.9

41.

120

NM

80U

216.0

4-0

.50

216.

0621

5.32

92.8

81.

183

164.

931.

9316

4.94

166.

9994

.30

1.18

1N

M80C

211.1

07.

1821

1.00

209.

1093

.14

1.14

916

0.19

4.91

160.

1316

1.82

94.2

01.

145

NM

90U

222.7

92.

1722

2.80

224.

5493

.04

1.23

417

2.99

4.65

172.

9417

4.13

94.4

81.

232

NM

90C

215.4

09.

4921

5.22

213.

7793

.34

1.17

516

6.88

5.43

166.

8116

5.08

93.7

41.

168

61

Tab

leE

2:In

fere

nce

for

the

mea

nb

ased

onfu

lly

mas

ked

Exp

onen

tial

(mea

n=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103×

103×

103

%L

en.

CD

100.

44-1

.62

100.

4499

.84

94.1

61.

000

71.3

90.

4171

.40

70.7

494

.22

1.00

0M

I106.

4418

.91

104.

7510

4.94

95.3

41.

051

74.3

210

.71

73.5

573

.24

95.0

01.

035

NM

10U

100.

67-1

.78

100.

6610

0.15

94.3

41.

003

71.6

00.

3971

.61

70.9

794

.02

1.00

3N

M10

C100.

93-1

.47

100.

9310

0.18

94.2

61.

003

71.6

60.

3671

.66

70.9

794

.38

1.00

3N

M20

U101.

54-1

.31

101.

5410

1.15

94.3

21.

013

72.2

30.

4572

.24

71.6

694

.64

1.01

3N

M20

C101.

68-1

.17

101.

6910

1.16

94.5

61.

013

72.1

80.

6072

.19

71.6

694

.40

1.01

3N

M30

U103.

20-1

.55

103.

2010

2.66

94.7

81.

028

73.0

40.

4573

.05

72.7

494

.66

1.02

8N

M30

C103.

41-0

.76

103.

4210

2.69

94.5

21.

029

72.9

80.

5972

.99

72.7

294

.50

1.02

8N

M40

U106.

26-0

.71

106.

2710

4.79

94.2

81.

050

75.0

00.

9375

.00

74.2

394

.66

1.04

9N

U40C

104.

83-0

.43

104.

8410

4.62

94.3

61.

048

74.6

00.

4574

.61

74.0

694

.54

1.04

7N

M50

U108.

85-1

.29

108.

8510

7.26

94.2

41.

074

75.6

70.

5475

.67

75.9

894

.76

1.07

4N

M50

C107.

64-0

.68

107.

6510

6.78

93.8

81.

070

76.1

71.

3876

.17

75.6

894

.56

1.07

0N

M60

U111.

01-0

.02

111.

0211

0.38

94.4

21.

106

78.3

91.

1378

.39

78.1

394

.70

1.10

4N

M60

C108.

43-0

.35

108.

4410

9.15

94.2

61.

093

77.8

41.

8377

.83

77.3

994

.68

1.09

4N

M70

U114.

37-0

.62

114.

3811

3.76

94.0

81.

139

81.2

21.

3681

.21

80.5

894

.58

1.13

9N

M70

C110.

821.

0911

0.83

111.

6994

.90

1.11

980

.03

1.64

80.0

279

.05

94.4

81.

118

NM

80U

118.

160.

1211

8.17

117.

8494

.26

1.18

082

.96

1.43

82.9

683

.40

94.8

21.

179

NM

80C

113.

601.

3911

3.60

114.

0894

.76

1.14

381

.89

1.78

81.8

880

.72

94.0

41.

141

NM

90U

124.

63-0

.08

124.

6412

2.47

94.1

41.

227

87.0

52.

2087

.03

86.7

694

.76

1.22

6N

M90

C117.

321.

8711

7.32

116.

3594

.46

1.16

583

.89

1.90

83.8

882

.31

94.6

01.

164

62

Tab

leE

3:In

fere

nce

for

the

mea

nb

ased

onfu

lly

mas

ked

Exp

onen

tial

(mea

n=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

31.0

5-1

.13

31.0

331

.59

95.1

21.

000

22.6

1-0

.15

22.6

122

.36

94.7

61.

000

MI

31.6

90.

7831

.69

32.3

195

.50

1.02

323

.12

0.93

23.1

022

.84

94.9

21.

021

NM

10U

31.1

5-1

.08

31.1

331

.69

95.0

61.

003

22.6

9-0

.17

22.6

922

.43

94.6

81.

003

NM

10C

31.1

7-1

.14

31.1

531

.69

95.1

41.

003

22.7

5-0

.15

22.7

522

.43

94.7

41.

003

NM

20U

31.4

0-1

.16

31.3

931

.99

95.2

41.

013

22.8

9-0

.06

22.8

922

.65

94.8

61.

013

NM

20C

31.4

0-0

.98

31.3

932

.00

95.0

81.

013

22.8

4-0

.15

22.8

422

.65

94.8

21.

013

NM

30U

31.9

1-1

.21

31.8

932

.48

95.2

21.

028

23.2

8-0

.28

23.2

822

.99

94.5

41.

028

NM

30C

32.0

3-0

.97

32.0

232

.47

95.2

21.

028

23.1

8-0

.16

23.1

822

.98

94.6

61.

028

NM

40U

32.8

1-1

.09

32.7

933

.13

94.8

61.

049

23.6

5-0

.09

23.6

623

.45

94.9

01.

049

NU

40C

32.6

0-1

.19

32.5

833

.07

95.4

81.

047

23.6

1-0

.10

23.6

123

.41

94.7

41.

047

NM

50U

33.3

4-1

.18

33.3

233

.92

95.3

61.

074

24.2

7-0

.41

24.2

724

.00

94.7

41.

074

NM

50C

32.8

2-1

.20

32.8

033

.76

95.2

61.

069

24.1

7-0

.06

24.1

723

.90

94.6

21.

069

NM

60U

34.5

8-0

.85

34.5

734

.87

94.9

81.

104

24.7

8-0

.05

24.7

824

.67

94.5

41.

104

NM

60C

34.1

0-0

.82

34.1

034

.52

95.1

81.

093

24.6

6-0

.28

24.6

624

.42

94.5

21.

092

NM

70U

35.8

7-1

.04

35.8

635

.95

95.0

21.

138

25.5

5-0

.24

25.5

525

.44

94.7

61.

138

NM

70C

34.4

3-1

.38

34.4

035

.25

95.2

61.

116

24.9

5-0

.26

24.9

524

.96

95.1

21.

116

NM

80U

36.8

1-0

.70

36.8

137

.21

95.1

21.

178

26.3

2-0

.21

26.3

326

.33

95.0

01.

178

NM

80C

35.6

1-0

.93

35.6

036

.01

94.8

81.

140

25.8

40.

1125

.84

25.4

994

.66

1.14

0N

M90U

38.0

3-1

.59

38.0

038

.65

95.2

21.

223

27.5

90.

0727

.59

27.3

794

.78

1.22

4N

M90C

36.4

1-1

.21

36.3

936

.71

95.4

81.

162

26.2

30.

1726

.23

25.9

994

.60

1.16

3

63

Tab

leE

4:In

fere

nce

for

the

log

scal

em

eanµ

bas

edfu

lly

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

180.0

8-0

.10

180.

0917

8.24

94.3

61.

000

142.

920.

9514

2.93

139.

2994

.06

1.00

0M

I18

4.6

5-0

.71

184.

6718

8.34

94.9

21.

057

145.

791.

1914

5.80

145.

0494

.64

1.04

1N

M10

U18

0.2

9-0

.32

180.

3017

8.55

94.1

81.

002

143.

210.

9014

3.22

139.

5194

.06

1.00

2N

M10

C18

0.4

6-0

.20

180.

4817

8.58

94.3

21.

002

143.

271.

0214

3.28

139.

5294

.14

1.00

2N

M20

U18

0.9

9-0

.37

181.

0017

9.34

93.9

81.

006

143.

301.

0714

3.31

140.

2494

.10

1.00

7N

M20

C18

1.3

4-0

.13

181.

3617

9.35

94.1

01.

006

143.

900.

8614

3.91

140.

2693

.90

1.00

7N

M30

U18

3.3

40.

3618

3.36

180.

9394

.20

1.01

514

4.94

1.68

144.

9514

1.52

94.2

21.

016

NM

30C

182.1

20.

7718

2.13

180.

8994

.42

1.01

514

5.26

1.37

145.

2714

1.35

93.8

81.

015

NM

40U

184.2

80.

4618

4.30

183.

2694

.32

1.02

814

5.78

1.85

145.

7814

3.30

94.4

21.

029

NM

40C

184.3

7-0

.38

184.

3918

2.93

94.1

61.

026

146.

271.

2814

6.28

142.

7894

.24

1.02

5N

M50

U18

9.3

8-0

.24

189.

4018

6.52

94.1

01.

046

149.

820.

8514

9.83

145.

6093

.96

1.04

5N

M50

C18

8.8

8-0

.51

188.

9018

5.02

94.4

01.

038

149.

541.

3114

9.55

144.

8793

.80

1.04

0N

M60

U19

3.1

4-0

.34

193.

1519

0.76

94.2

41.

070

152.

491.

3715

2.50

149.

1493

.82

1.07

1N

M60

C18

9.6

10.

4318

9.63

188.

2194

.10

1.05

615

1.02

0.31

151.

0314

7.18

93.9

61.

057

NM

70U

199.3

41.

1319

9.36

196.

1793

.90

1.10

115

9.06

-0.0

015

9.07

153.

7893

.44

1.10

4N

M70

C19

5.0

7-1

.62

195.

0819

1.19

93.8

81.

073

153.

841.

0015

3.85

149.

6793

.90

1.07

5N

M80

U20

7.7

12.

0820

7.72

204.

1493

.90

1.14

516

5.26

0.68

165.

2815

9.88

94.0

61.

148

NM

80C

199.2

20.

0919

9.24

194.

3693

.42

1.09

015

7.14

0.27

157.

1615

2.23

93.9

21.

093

NM

90U

223.6

83.

9822

3.67

217.

4193

.98

1.22

017

2.53

2.11

172.

5417

0.03

94.0

61.

221

NM

90C

201.5

71.

4820

1.58

198.

5694

.04

1.11

415

9.67

1.63

159.

6815

4.99

93.5

61.

113

64

Tab

leE

5:In

fere

nce

for

the

log

scal

em

eanµ

bas

edfu

lly

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

101.6

00.

8410

1.60

99.2

794

.00

1.00

070

.05

0.68

70.0

670

.51

95.1

01.

000

MI

104.0

50.

7510

4.05

102.

2794

.26

1.03

071

.57

0.79

71.5

772

.30

95.2

41.

025

NM

10U

101.7

10.

9210

1.71

99.4

594

.22

1.00

270

.15

0.61

70.1

570

.63

95.2

21.

002

NM

10C

101.8

01.

0510

1.81

99.4

394

.22

1.00

270

.21

0.66

70.2

170

.64

95.0

81.

002

NM

20U

102.5

40.

9810

2.55

99.9

694

.32

1.00

770

.43

0.62

70.4

470

.98

95.1

41.

007

NM

20C

102.2

31.

0410

2.23

99.9

594

.20

1.00

770

.54

0.62

70.5

470

.99

95.1

81.

007

NM

30U

102.9

00.

5510

2.91

100.

7794

.24

1.01

571

.00

0.42

71.0

171

.61

95.2

41.

016

NM

30C

102.8

30.

9010

2.83

100.

7194

.34

1.01

471

.07

0.82

71.0

771

.55

95.3

01.

015

NM

40U

104.3

31.

0010

4.34

102.

0794

.32

1.02

872

.49

0.57

72.5

072

.53

95.2

41.

029

NM

40C

103.4

50.

7810

3.46

101.

8094

.12

1.02

571

.89

0.24

71.9

072

.36

95.4

21.

026

NM

50U

106.5

51.

2810

6.56

103.

8794

.12

1.04

673

.22

0.62

73.2

273

.76

95.1

61.

046

NM

50C

105.1

20.

5910

5.13

103.

1394

.08

1.03

973

.05

0.65

73.0

673

.31

95.1

21.

040

NM

60U

108.2

02.

0310

8.19

106.

2794

.24

1.07

075

.41

0.55

75.4

175

.44

95.1

61.

070

NM

60C

107.5

20.

5010

7.53

104.

8194

.08

1.05

674

.51

0.08

74.5

174

.38

94.8

41.

055

NM

70U

111.4

8-0

.52

111.

4910

9.50

94.0

61.

103

78.3

21.

1778

.32

77.7

994

.66

1.10

3N

M70C

109.2

10.

7910

9.22

106.

4994

.14

1.07

375

.00

0.53

75.0

175

.63

95.2

81.

073

NM

80U

116.5

70.

6311

6.58

114.

0593

.90

1.14

981

.71

0.26

81.7

281

.00

94.8

81.

149

NM

80C

110.1

50.

4611

0.16

108.

4494

.50

1.09

276

.57

1.14

76.5

776

.95

94.7

21.

091

NM

90U

123.5

21.

2412

3.52

121.

1494

.56

1.22

085

.86

-0.1

685

.87

86.0

595

.00

1.22

0N

M90C

112.4

10.

3111

2.42

110.

6394

.26

1.11

477

.98

0.74

77.9

878

.44

95.0

01.

112

65

Tab

leE

6:In

fere

nce

for

the

log

scal

em

eanµ

bas

edfu

lly

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

31.5

80.

5931

.58

31.6

095

.20

1.00

022

.55

0.24

22.5

522

.36

94.8

41.

000

MI

32.0

20.

5432

.02

32.2

695

.10

1.02

123

.03

0.27

23.0

322

.81

94.9

41.

020

NM

10U

31.6

30.

6131

.63

31.6

695

.08

1.00

222

.60

0.26

22.6

022

.40

94.8

21.

002

NM

10C

31.6

30.

6031

.63

31.6

695

.02

1.00

222

.56

0.26

22.5

622

.40

94.9

61.

002

NM

20U

31.8

70.

5131

.87

31.8

295

.20

1.00

722

.70

0.21

22.7

022

.51

94.9

81.

007

NM

20C

31.9

10.

6431

.91

31.8

195

.00

1.00

722

.69

0.20

22.6

922

.51

94.9

01.

007

NM

30U

32.1

30.

6432

.12

32.0

995

.02

1.01

522

.84

0.17

22.8

422

.71

95.0

41.

016

NM

30C

32.1

00.

8032

.09

32.0

795

.28

1.01

522

.86

0.28

22.8

622

.69

94.8

41.

015

NM

40U

32.5

60.

6132

.56

32.5

094

.96

1.02

823

.10

0.16

23.1

123

.00

94.9

01.

028

NM

40C

32.3

10.

6932

.30

32.4

195

.14

1.02

623

.06

0.30

23.0

622

.93

94.9

21.

026

NM

50U

32.9

60.

5632

.95

33.0

895

.26

1.04

723

.53

0.27

23.5

323

.40

94.5

41.

046

NM

50C

32.8

20.

6032

.82

32.8

495

.00

1.03

923

.49

0.21

23.4

923

.24

95.2

21.

039

NM

60U

33.7

50.

7033

.75

33.8

395

.00

1.07

024

.00

0.20

24.0

023

.93

95.1

21.

070

NM

60C

33.2

30.

4433

.24

33.3

595

.10

1.05

523

.65

0.12

23.6

523

.59

94.9

61.

055

NM

70U

34.5

40.

3634

.54

34.8

695

.34

1.10

324

.94

0.04

24.9

424

.66

95.0

41.

103

NM

70C

34.1

50.

6034

.15

33.9

295

.00

1.07

324

.31

-0.0

124

.31

23.9

994

.68

1.07

3N

M80U

36.6

20.

8036

.61

36.3

294

.52

1.14

925

.52

0.42

25.5

225

.69

95.2

81.

149

NM

80C

34.8

00.

5334

.80

34.5

394

.50

1.09

324

.59

0.09

24.6

024

.42

95.0

21.

092

NM

90U

38.5

20.

4938

.52

38.5

594

.94

1.22

027

.60

0.09

27.6

027

.27

94.6

81.

220

NM

90C

35.0

60.

4935

.06

35.1

894

.96

1.11

324

.94

0.43

24.9

424

.89

94.6

81.

113

66

Tab

leE

7:In

fere

nce

for

the

log

scal

eva

rian

ceσ

2b

ased

onfu

lly

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

257.5

0-3

0.14

255.

7625

0.42

89.6

01.

000

199.

30-1

9.91

198.

3219

6.02

91.4

81.

000

MI

283.0

540

.68

280.

1429

3.92

94.1

41.

174

211.

9521

.45

210.

8921

6.95

94.2

41.

107

NM

10U

258.0

2-3

0.13

256.

2825

1.28

89.5

41.

003

200.

32-2

0.09

199.

3319

6.65

91.3

41.

003

NM

10C

257.8

2-2

9.83

256.

1225

1.36

89.4

81.

004

200.

29-1

9.97

199.

3119

6.67

91.3

21.

003

NM

20U

260.0

3-3

1.82

258.

1025

3.48

89.2

01.

012

202.

43-2

0.00

201.

4619

8.71

91.4

61.

014

NM

20C

261.9

7-3

1.11

260.

1525

3.59

89.7

21.

013

201.

96-1

9.44

201.

0419

8.76

91.5

81.

014

NM

30U

264.2

1-3

2.18

262.

2725

7.93

89.6

41.

030

206.

21-1

9.50

205.

3020

2.32

91.5

81.

032

NM

30C

266.4

0-3

0.53

264.

6825

7.95

89.5

61.

030

204.

69-2

0.33

203.

7020

1.85

91.5

61.

030

NM

40U

272.1

0-3

2.54

270.

1726

4.54

89.3

81.

056

210.

89-2

0.43

209.

9220

7.30

91.5

21.

058

NM

40C

272.0

7-3

0.22

270.

4126

3.81

89.4

81.

053

209.

70-2

2.07

208.

5520

5.98

91.4

81.

051

NM

50U

279.6

7-3

2.89

277.

7527

3.75

90.1

41.

093

215.

93-2

4.10

214.

6121

3.73

91.6

01.

090

NM

50C

279.8

5-3

4.70

277.

7226

9.91

89.1

01.

078

213.

91-2

0.04

213.

0021

2.00

91.8

81.

082

NM

60U

299.2

1-3

4.27

297.

2728

5.98

88.9

61.

142

227.

60-2

2.26

226.

5422

3.82

91.6

21.

142

NM

60C

286.5

5-3

2.00

284.

7827

9.20

89.2

41.

115

221.

25-1

9.18

220.

4421

8.83

91.7

61.

116

NM

70U

309.4

4-4

2.62

306.

5330

0.97

89.4

01.

202

238.

43-2

2.68

237.

3723

7.02

91.5

21.

209

NM

70C

298.1

4-3

5.21

296.

0928

8.17

89.6

01.

151

230.

41-1

9.67

229.

5922

6.32

91.5

01.

155

NM

80U

334.6

6-4

9.35

331.

0332

2.94

88.1

01.

290

260.

19-3

0.19

258.

4625

3.81

90.7

21.

295

NM

80C

305.8

5-4

0.30

303.

2229

7.72

89.5

61.

189

238.

91-2

2.74

237.

8523

4.13

91.3

21.

194

NM

90U

366.8

2-5

2.33

363.

1035

6.45

88.3

81.

423

286.

05-3

0.25

284.

4828

0.63

91.3

01.

432

NM

90C

317.9

5-3

5.60

315.

9831

0.72

89.2

21.

241

247.

29-2

5.57

245.

9824

2.69

91.4

81.

238

67

Tab

leE

8:In

fere

nce

for

the

log

scal

eva

rian

ceσ

2b

ased

onfu

lly

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

138

.01

-9.7

213

7.68

140.

0594

.04

1.00

010

2.64

-2.9

010

2.61

99.7

193

.36

1.00

0M

I144

.13

10.6

414

3.75

148.

7195

.28

1.06

210

6.04

7.62

105.

7810

3.76

94.0

81.

041

NM

10U

138

.46

-9.5

913

8.15

140.

5493

.74

1.00

410

2.85

-2.8

710

2.82

100.

0593

.42

1.00

3N

M10C

138

.56

-9.9

213

8.22

140.

4994

.06

1.00

310

2.97

-2.7

910

2.94

100.

0593

.36

1.00

3N

M20U

140

.11

-9.5

313

9.80

141.

9993

.84

1.01

410

4.11

-3.2

510

4.07

101.

0393

.18

1.01

3N

M20C

139

.62

-9.4

713

9.31

141.

9593

.90

1.01

410

3.74

-2.7

810

3.71

101.

0593

.58

1.01

3N

M30U

143

.65

-10.

8614

3.26

144.

2893

.50

1.03

010

6.28

-2.8

710

6.26

102.

8293

.36

1.03

1N

M30C

142

.46

-10.

3814

2.10

144.

1393

.44

1.02

910

6.02

-2.8

210

5.99

102.

6793

.20

1.03

0N

M40U

146

.98

-10.

8314

6.59

147.

9493

.68

1.05

610

8.47

-2.9

310

8.44

105.

4093

.62

1.05

7N

M40C

145

.02

-10.

6014

4.65

147.

2793

.76

1.05

210

7.08

-1.9

010

7.07

105.

0193

.66

1.05

3N

M50U

151

.89

-10.

5115

1.54

153.

0593

.60

1.09

311

1.76

-3.9

211

1.71

108.

8793

.58

1.09

2N

M50C

149

.72

-11.

3414

9.31

151.

1493

.62

1.07

911

1.15

-2.4

111

1.14

107.

7694

.04

1.08

1N

M60U

159

.34

-10.

9115

8.98

159.

8593

.38

1.14

111

7.71

-4.5

211

7.63

113.

6593

.50

1.14

0N

M60C

155

.28

-9.4

315

5.01

156.

1193

.80

1.11

511

3.63

-3.8

211

3.57

110.

9593

.58

1.11

3N

M70U

169

.59

-11.

8816

9.19

169.

0593

.00

1.20

712

3.41

-3.6

312

3.37

120.

3393

.84

1.20

7N

M70C

163

.02

-11.

6016

2.62

161.

1893

.28

1.15

111

7.71

-4.2

211

7.64

114.

7193

.28

1.15

0N

M80U

182

.80

-13.

7218

2.31

181.

8193

.10

1.29

813

1.21

-5.2

513

1.11

129.

3493

.40

1.29

7N

M80C

166

.68

-11.

5416

6.29

167.

1493

.24

1.19

312

1.30

-6.1

612

1.16

118.

7293

.28

1.19

1N

M90U

201

.43

-15.

0620

0.89

201.

0592

.90

1.43

614

8.31

-3.7

214

8.27

143.

2092

.94

1.43

6N

M90C

170

.98

-9.2

417

0.75

173.

9293

.86

1.24

212

6.36

-5.1

312

6.27

123.

3994

.04

1.23

7

68

Tab

leE

9:In

fere

nce

for

the

log

scal

eva

rian

ceσ

2b

ased

onfu

lly

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

45.0

4-0

.63

45.0

444

.69

94.6

41.

000

32.1

70.

2032

.17

31.6

394

.86

1.00

0M

I46.0

81.

4646

.06

45.7

794

.72

1.02

433

.06

1.20

33.0

432

.32

94.7

01.

022

NM

10U

45.2

3-0

.62

45.2

344

.84

94.6

41.

003

32.2

80.

2132

.28

31.7

494

.74

1.00

3N

M10C

45.2

1-0

.54

45.2

144

.85

94.6

61.

003

32.2

70.

2032

.28

31.7

394

.76

1.00

3N

M20U

45.4

5-0

.51

45.4

545

.30

94.6

61.

014

32.3

90.

2132

.40

32.0

695

.02

1.01

4N

M20C

45.5

9-0

.63

45.5

945

.29

94.9

01.

013

32.6

10.

2632

.61

32.0

594

.82

1.01

3N

M30U

46.3

2-0

.87

46.3

246

.07

94.8

81.

031

32.9

50.

1232

.96

32.6

194

.74

1.03

1N

M30C

46.3

7-0

.55

46.3

746

.02

94.8

21.

030

32.9

20.

3032

.92

32.5

794

.62

1.03

0N

M40U

47.7

8-0

.79

47.7

847

.23

94.1

81.

057

33.8

20.

0933

.82

33.4

294

.74

1.05

7N

M40C

47.2

0-0

.75

47.2

047

.01

94.6

21.

052

33.6

70.

0733

.67

33.2

794

.98

1.05

2N

M50U

49.2

9-0

.06

49.3

048

.86

94.4

01.

093

34.8

30.

4934

.83

34.5

794

.88

1.09

3N

M50C

48.3

8-0

.99

48.3

848

.26

94.4

61.

080

34.6

60.

6734

.66

34.1

894

.82

1.08

0N

M60U

51.7

5-0

.95

51.7

450

.98

94.1

01.

141

36.5

20.

2536

.52

36.0

994

.68

1.14

1N

M60C

50.0

0-0

.82

50.0

049

.75

94.8

41.

113

35.8

1-0

.03

35.8

135

.21

95.1

41.

113

NM

70U

54.5

2-1

.42

54.5

053

.90

94.3

81.

206

38.4

6-0

.06

38.4

738

.16

94.7

81.

206

NM

70C

51.5

3-0

.29

51.5

451

.47

95.2

01.

152

36.9

0-0

.02

36.9

136

.41

94.3

41.

151

NM

80U

58.5

0-0

.96

58.4

958

.04

94.9

01.

299

41.7

6-0

.25

41.7

741

.06

94.5

81.

298

NM

80C

53.9

9-0

.31

54.0

053

.36

94.4

21.

194

37.8

40.

0537

.84

37.7

494

.88

1.19

3N

M90U

64.1

6-1

.05

64.1

664

.17

94.7

01.

436

45.9

80.

2845

.98

45.4

194

.86

1.43

6N

M90C

55.7

8-0

.53

55.7

855

.39

94.7

01.

239

40.0

70.

3140

.07

39.1

994

.24

1.23

9

69

Tab

leE

10:

Infe

ren

cefo

rth

em

eaneµ

+σ2/2

bas

edon

full

ym

aske

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(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103

×10

3×

103

×10

3%

Len

.

CD

378.1

815

.85

377.

8837

1.76

91.0

01.

000

297.

2510

.78

297.

0828

6.92

92.3

21.

000

MI

527.6

719

6.00

489.

9760

0.18

96.1

81.

614

353.

9110

6.84

337.

4336

2.65

96.0

41.

264

NM

10U

378.8

715

.63

378.

5937

2.58

91.0

21.

002

298.

1410

.71

297.

9828

7.56

92.3

21.

002

NM

10C

379.5

516

.18

379.

2537

2.79

91.0

01.

003

298.

3511

.03

298.

1828

7.63

92.3

81.

002

NM

20U

380.2

814

.53

380.

0437

4.54

91.0

81.

007

299.

5411

.30

299.

3628

9.70

92.3

21.

010

NM

20C

384.0

316

.13

383.

7337

5.23

90.8

61.

009

300.

2611

.50

300.

0728

9.78

92.3

41.

010

NM

30U

386.0

116

.58

385.

6937

9.45

90.8

01.

021

304.

9013

.54

304.

6229

3.65

92.4

21.

023

NM

30C

384.5

218

.22

384.

1338

0.01

91.2

41.

022

303.

3312

.14

303.

1129

2.95

92.3

21.

021

NM

40U

391.6

017

.42

391.

2538

6.11

91.0

41.

039

305.

8613

.22

305.

6129

8.20

92.8

61.

039

NM

40C

392.6

818

.22

392.

3038

6.10

91.0

01.

039

307.

5111

.30

307.

3429

6.95

92.2

01.

035

NM

50U

403.9

418

.74

403.

5539

5.03

90.6

41.

063

315.

7810

.44

315.

6430

3.57

91.9

41.

058

NM

50C

403.7

616

.76

403.

4539

2.13

90.8

61.

055

310.

9613

.63

310.

6930

3.14

92.6

61.

057

NM

60U

416.5

820

.13

416.

1340

6.48

90.4

81.

093

323.

9414

.09

323.

6731

2.95

92.0

61.

091

NM

60C

408.7

421

.47

408.

2240

2.53

91.1

61.

083

322.

0914

.44

321.

7931

0.22

92.4

01.

081

NM

70U

431.5

918

.93

431.

2241

7.67

90.3

21.

123

333.

4013

.32

333.

1632

2.92

91.6

61.

125

NM

70C

425.2

019

.30

424.

8141

1.56

90.2

21.

107

332.

2216

.97

331.

8231

8.19

92.1

21.

109

NM

80U

440.4

117

.28

440.

1143

1.57

89.8

41.

161

349.

0411

.19

348.

9033

3.76

91.3

61.

163

NM

80C

429.3

519

.09

428.

9742

0.70

90.0

81.

132

340.

8014

.88

340.

5132

5.47

91.9

01.

134

NM

90U

466.6

124

.08

466.

0445

1.88

89.2

61.

216

357.

6714

.97

357.

3934

8.83

91.5

81.

216

NM

90C

448.0

628

.09

447.

2243

6.24

91.1

81.

173

349.

1016

.19

348.

7633

4.07

91.7

81.

164

70

Tab

leE

11:

Infe

ren

cefo

rth

em

eaneµ

+σ2/2

bas

edon

full

ym

aske

dLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

204.5

15.

8620

4.44

202.

4193

.66

1.00

014

2.03

4.77

141.

9614

3.32

94.9

01.

000

MI

224.1

949

.82

218.

6122

6.79

95.7

01.

120

150.

9826

.74

148.

6115

2.96

95.8

61.

067

NM

10U

204.8

66.

1420

4.79

202.

9293

.76

1.00

314

2.30

4.70

142.

2414

3.64

95.0

21.

002

NM

10C

204.9

56.

0920

4.88

202.

8793

.64

1.00

214

2.57

4.88

142.

5014

3.66

94.8

01.

002

NM

20U

206.8

86.

5220

6.80

204.

3793

.64

1.01

014

3.59

4.52

143.

5414

4.56

94.9

01.

009

NM

20C

206.0

56.

5820

5.96

204.

3593

.84

1.01

014

3.09

4.87

143.

0214

4.61

94.7

21.

009

NM

30U

208.4

14.

9220

8.37

206.

3693

.44

1.02

014

5.38

4.65

145.

3214

6.26

94.6

01.

021

NM

30C

207.6

45.

7720

7.59

206.

3693

.62

1.02

014

5.10

5.32

145.

0214

6.22

94.8

61.

020

NM

40U

212.8

86.

2121

2.81

209.

9293

.56

1.03

714

7.66

5.06

147.

5914

8.65

94.8

61.

037

NM

40C

210.6

55.

7821

0.59

209.

4193

.78

1.03

514

6.45

5.24

146.

3714

8.43

94.9

01.

036

NM

50U

216.8

77.

4521

6.76

214.

6093

.56

1.06

014

9.63

4.49

149.

5815

1.65

94.5

81.

058

NM

50C

215.2

85.

4221

5.23

213.

1193

.74

1.05

315

1.46

5.94

151.

3615

1.16

94.6

61.

055

NM

60U

221.7

18.

9322

1.55

220.

4993

.72

1.08

915

5.35

4.40

155.

3115

5.61

94.6

81.

086

NM

60C

220.4

47.

4822

0.34

218.

1593

.80

1.07

815

2.95

4.00

152.

9115

3.99

94.7

61.

074

NM

70U

230.4

75.

1623

0.43

227.

0593

.40

1.12

216

1.69

6.76

161.

5716

0.92

94.6

81.

123

NM

70C

228.0

47.

1522

7.95

223.

0493

.50

1.10

215

6.69

4.72

156.

6415

7.62

94.5

21.

100

NM

80U

241.3

37.

0524

1.26

235.

7793

.06

1.16

516

7.20

4.46

167.

1616

6.59

94.3

01.

162

NM

80C

230.2

56.

9923

0.16

228.

6593

.30

1.13

016

0.87

4.55

160.

8316

1.39

94.3

81.

126

NM

90U

249.8

38.

0124

9.72

245.

7093

.44

1.21

417

3.55

5.71

173.

4717

3.94

94.3

61.

214

NM

90C

235.4

89.

3823

5.32

235.

3893

.56

1.16

316

3.84

5.02

163.

7816

5.80

94.4

41.

157

71

Tab

leE

12:

Infe

ren

cefo

rth

em

eaneµ

+σ2/2

bas

edon

full

ym

aske

dLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

63.7

21.

6963

.70

63.9

395

.28

1.00

046

.10

1.20

46.0

945

.20

94.5

21.

000

MI

65.3

25.

8265

.06

65.7

795

.60

1.02

947

.32

3.33

47.2

046

.30

94.4

61.

024

NM

10U

63.8

81.

7363

.87

64.0

795

.30

1.00

246

.22

1.24

46.2

145

.31

94.4

41.

002

NM

10C

63.8

61.

7963

.85

64.0

895

.30

1.00

246

.11

1.24

46.1

045

.31

94.5

41.

002

NM

20U

64.2

11.

6664

.20

64.5

195

.22

1.00

946

.33

1.17

46.3

245

.61

94.7

21.

009

NM

20C

64.3

31.

7964

.31

64.5

095

.12

1.00

946

.56

1.20

46.5

545

.61

94.5

81.

009

NM

30U

64.9

31.

6164

.92

65.2

295

.36

1.02

046

.88

1.04

46.8

846

.12

94.4

01.

020

NM

30C

65.2

02.

1465

.18

65.2

195

.10

1.02

046

.89

1.38

46.8

746

.10

94.3

81.

020

NM

40U

66.2

51.

6966

.24

66.2

795

.14

1.03

747

.72

1.03

47.7

146

.86

94.3

41.

037

NM

40C

65.9

21.

8465

.90

66.1

495

.32

1.03

547

.70

1.24

47.6

946

.76

94.4

01.

034

NM

50U

67.4

52.

2667

.42

67.7

295

.12

1.05

948

.52

1.56

48.5

047

.87

94.5

21.

059

NM

50C

66.5

81.

5266

.57

67.3

095

.42

1.05

348

.59

1.61

48.5

747

.63

94.5

61.

054

NM

60U

69.3

11.

8269

.29

69.4

495

.04

1.08

650

.04

1.29

50.0

349

.11

94.1

21.

086

NM

60C

67.8

41.

4467

.84

68.7

195

.20

1.07

549

.30

0.91

49.2

948

.58

94.3

01.

075

NM

70U

70.6

70.

9370

.67

71.6

094

.98

1.12

051

.51

0.82

51.5

150

.65

94.5

81.

120

NM

70C

70.3

62.

2470

.33

70.3

594

.86

1.10

050

.79

0.75

50.7

949

.69

94.6

41.

099

NM

80U

73.9

92.

1773

.96

74.3

694

.98

1.16

352

.87

1.33

52.8

652

.56

94.7

01.

163

NM

80C

72.4

22.

2072

.40

72.1

095

.06

1.12

851

.78

1.00

51.7

750

.94

94.5

61.

127

NM

90U

77.3

31.

7577

.32

77.4

694

.86

1.21

255

.72

1.32

55.7

154

.78

94.5

01.

212

NM

90C

74.0

12.

0373

.99

73.9

895

.36

1.15

753

.21

1.82

53.1

952

.31

94.3

81.

157

72

Tab

leE

13:

Infe

ren

cefo

rth

eva

rian

cee2µ

+2σ2−e2µ

+σ2

bas

edon

fully

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

5464.

0699

0.0

453

74.1

647

24.9

482

.26

1.00

036

16.2

656

2.15

3572

.65

3250

.99

85.6

61.

000

MI

2×

1010

3×

108

2×

1010

2×

1010

97.8

23×

106

2221

1.99

5685

.50

2147

4.17

2744

4.25

96.8

88.

442

NM

10U

5484.

6799

4.0

253

94.3

847

43.1

482

.32

1.00

436

35.9

856

6.53

3591

.93

3264

.36

85.6

41.

004

NM

10C

5489.

3510

00.5

553

97.9

347

47.8

082

.10

1.00

536

46.0

257

0.21

3601

.51

3267

.23

85.7

01.

005

NM

20U

5495.

0498

5.6

054

06.4

747

74.1

881

.48

1.01

036

85.5

758

4.45

3639

.30

3307

.95

85.8

01.

018

NM

20C

5648.

2710

32.9

655

53.5

748

27.6

481

.70

1.02

237

04.5

458

9.24

3657

.74

3310

.79

85.7

81.

018

NM

30U

5723.

7210

41.1

156

28.8

049

06.5

081

.82

1.03

838

36.2

863

3.01

3784

.08

3396

.00

85.5

81.

045

NM

30C

5765.

7110

74.6

756

65.2

349

41.1

882

.04

1.04

637

44.2

060

2.28

3695

.81

3366

.48

85.8

61.

036

NM

40U

6015.

4911

10.9

059

12.6

150

91.8

481

.50

1.07

838

51.5

763

8.69

3798

.62

3473

.16

85.4

21.

068

NM

40C

6062.

5011

46.3

959

53.7

351

16.3

381

.60

1.08

338

43.9

761

6.21

3794

.64

3443

.65

85.3

21.

059

NM

50U

6268.

3111

91.1

461

54.7

153

17.0

281

.68

1.12

539

94.1

263

8.87

3943

.09

3570

.82

84.7

61.

098

NM

50C

6301.

0511

80.1

661

90.1

652

74.8

080

.80

1.11

639

29.8

166

2.51

3873

.95

3560

.72

85.6

01.

095

NM

60U

7353.

5113

98.1

072

20.1

057

94.8

480

.92

1.22

642

82.0

074

7.65

4216

.64

3797

.12

84.8

61.

168

NM

60C

6682.

9613

02.5

065

55.4

655

55.3

681

.52

1.17

642

53.7

675

1.44

4187

.28

3736

.56

85.4

61.

149

NM

70U

7634.

4914

35.3

574

99.1

060

88.8

080

.18

1.28

946

07.5

780

0.36

4537

.98

4018

.59

84.6

21.

236

NM

70C

7082.

9814

11.9

269

41.5

258

55.7

880

.70

1.23

946

41.1

284

0.43

4564

.84

3928

.95

84.9

21.

209

NM

80U

8395.

0515

22.4

282

56.6

865

65.2

379

.70

1.38

951

46.5

588

4.66

5070

.46

4341

.52

83.4

81.

335

NM

80C

7470.

5114

15.9

373

35.8

360

46.7

881

.14

1.28

049

56.7

886

9.69

4880

.38

4086

.03

84.6

21.

257

NM

90U

9161.

2818

30.4

589

77.4

574

67.5

977

.96

1.58

056

60.2

410

41.2

855

64.1

948

26.7

882

.72

1.48

5N

M90C

8440.

4117

07.3

382

66.7

566

10.9

681

.54

1.39

951

05.7

991

7.40

5023

.20

4264

.82

83.6

01.

312

73

Tab

leE

14:

Infe

ren

cefo

rth

eva

rian

cee2µ

+2σ2−e2µ

+σ2

bas

edon

fully

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103

×10

3×

103

×10

3%

Len

.

CD

2159

.09

254

.04

2144

.30

2101

.53

90.1

21.

000

1487

.10

157.

2214

78.9

114

42.3

791

.88

1.00

0M

I3536

.33

1627

.36

3139

.95

4214

.81

96.9

42.

006

1927

.71

739.

4517

80.4

219

76.8

196

.64

1.37

1N

M10

U2166

.99

258

.30

2151

.75

2109

.92

89.8

81.

004

1492

.85

157.

8514

84.6

314

46.9

192

.12

1.00

3N

M10

C2167

.72

255

.71

2152

.80

2108

.46

90.0

81.

003

1494

.89

160.

0314

86.4

514

47.6

792

.12

1.00

4N

M20

U2196

.00

267

.78

2179

.83

2133

.84

90.1

41.

015

1509

.26

157.

8615

01.1

314

59.8

591

.84

1.01

2N

M20

C2176

.31

265

.75

2160

.24

2131

.94

90.3

01.

014

1500

.24

161.

5714

91.6

714

60.8

391

.94

1.01

3N

M30

U2240

.86

260

.87

2225

.85

2163

.70

89.6

61.

030

1543

.65

167.

6615

34.6

814

85.8

092

.18

1.03

0N

M30

C2232

.94

264

.55

2217

.44

2162

.65

89.6

21.

029

1535

.95

170.

3915

26.6

214

84.9

991

.96

1.03

0N

M40

U2305

.33

284

.43

2287

.94

2222

.59

89.8

41.

058

1574

.44

174.

9915

64.8

415

20.3

491

.98

1.05

4N

M40

C2277

.00

275

.55

2260

.49

2210

.26

90.0

81.

052

1559

.49

180.

1315

49.2

115

17.4

892

.50

1.05

2N

M50

U2389

.94

312

.17

2369

.70

2301

.67

88.9

61.

095

1604

.96

172.

1515

95.8

615

62.3

691

.56

1.08

3N

M50

C2370

.68

289

.06

2353

.23

2271

.32

89.1

61.

081

1645

.86

196.

2416

34.2

815

59.9

091

.76

1.08

1N

M60

U2507

.79

346

.20

2484

.03

2405

.16

89.2

41.

144

1697

.23

185.

7516

87.2

016

24.8

291

.38

1.12

6N

M60

C2482

.45

338

.15

2459

.56

2362

.17

89.5

81.

124

1640

.05

177.

8916

30.5

415

95.2

392

.00

1.10

6N

M70

U2659

.77

361

.82

2635

.31

2526

.98

87.6

41.

202

1784

.13

224.

4017

70.1

417

13.7

691

.44

1.18

8N

M70

C2621

.48

359

.44

2596

.98

2447

.88

88.6

01.

165

1715

.66

193.

6317

04.8

716

50.1

091

.62

1.14

4N

M80

U2905

.02

419

.25

2874

.89

2707

.91

87.7

01.

289

1881

.03

219.

6718

68.3

418

08.4

291

.30

1.25

4N

M80

C2644

.62

371

.44

2618

.67

2533

.97

88.4

41.

206

1777

.77

191.

5117

67.6

117

02.8

791

.34

1.18

1N

M90

U3159

.05

476

.65

3123

.19

2943

.67

87.2

81.

401

2062

.15

281.

5420

43.0

519

67.9

090

.22

1.36

4N

M90

C2735

.17

424

.22

2702

.34

2651

.43

88.7

81.

262

1826

.14

214.

6018

13.6

717

71.4

291

.18

1.22

8

74

Tab

leE

15:

Infe

ren

cefo

rth

eva

rian

cee2µ

+2σ2−e2µ

+σ2

bas

edon

fully

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

624.

7834

.30

623.

9062

1.18

94.6

01.

000

448.

3023

.69

447.

7243

7.80

94.3

81.

000

MI

669.

0813

4.94

655.

4066

8.20

95.9

21.

076

471.

5573

.37

465.

8545

8.41

94.8

21.

047

NM

10U

626.

8534

.78

625.

9562

3.09

94.5

81.

003

449.

8324

.11

449.

2343

9.14

94.4

61.

003

NM

10C

627.

0035

.76

626.

0462

3.25

94.6

01.

003

449.

1023

.90

448.

5143

9.11

94.2

01.

003

NM

20U

629.

9335

.50

628.

9962

8.79

94.6

41.

012

450.

9223

.71

450.

3544

3.02

94.6

21.

012

NM

20C

631.

7135

.47

630.

7862

8.63

94.4

41.

012

454.

1124

.54

453.

4944

3.03

94.6

41.

012

NM

30U

639.

3833

.49

638.

5763

7.90

94.3

81.

027

457.

8322

.83

457.

3144

9.59

94.6

21.

027

NM

30C

643.

4739

.21

642.

3463

8.18

94.5

61.

027

457.

7626

.08

457.

0644

9.51

94.6

01.

027

NM

40U

658.

3636

.42

657.

4265

2.12

94.3

61.

050

468.

9423

.42

468.

4045

9.32

93.9

41.

049

NM

40C

654.

2037

.16

653.

2165

0.17

94.7

61.

047

468.

2724

.33

467.

6945

7.97

94.3

01.

046

NM

50U

678.

5746

.95

677.

0167

2.48

94.5

81.

083

481.

1730

.13

480.

2747

3.15

94.3

81.

081

NM

50C

662.

7934

.43

661.

9666

5.15

94.8

01.

071

480.

9831

.74

479.

9846

9.74

94.5

01.

073

NM

60U

704.

4840

.71

703.

3869

5.62

94.2

61.

120

501.

2628

.43

500.

5049

0.02

94.4

61.

119

NM

60C

682.

7337

.28

681.

7868

3.97

94.5

21.

101

491.

6623

.57

491.

1448

1.64

94.1

41.

100

NM

70U

726.

7834

.95

726.

0172

5.96

93.9

61.

169

519.

3025

.01

518.

7551

1.64

94.5

81.

169

NM

70C

711.

4848

.50

709.

9070

6.76

95.1

21.

138

507.

7623

.97

507.

2549

6.39

94.1

61.

134

NM

80U

773.

6150

.04

772.

0776

9.36

94.4

81.

239

550.

3329

.01

549.

6254

0.72

94.3

01.

235

NM

80C

741.

4151

.57

739.

6973

0.52

94.6

21.

176

520.

1826

.95

519.

5451

3.00

94.6

61.

172

NM

90U

827.

0353

.58

825.

3782

3.99

93.8

61.

326

590.

6936

.25

589.

6357

9.55

94.0

61.

324

NM

90C

766.

0151

.88

764.

3375

5.69

94.2

21.

217

545.

6935

.58

544.

5953

1.74

94.3

21.

215

75

Tab

leE

16:

Infe

ren

cefo

rth

e84

thp

erce

nti

leeµ

+σ

bas

edon

fully

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103

×10

3×

103

×10

3%

Len

.

CD

614.8

12.

6661

4.87

604.

3790

.88

1.00

048

6.00

3.89

486.

0446

9.27

92.2

41.

000

MI

747.0

621

4.82

715.

5880

1.30

95.4

21.

326

545.

8112

4.34

531.

5155

3.13

95.3

41.

179

NM

10U

615.8

92.

2261

5.95

605.

6990

.80

1.00

248

7.47

3.63

487.

5147

0.28

92.0

81.

002

NM

10C

617.0

53.

1461

7.10

606.

0090

.90

1.00

348

7.74

4.17

487.

7747

0.39

92.2

01.

002

NM

20U

618.5

00.

0061

8.56

608.

9590

.98

1.00

848

9.57

4.31

489.

6047

3.69

92.0

81.

009

NM

20C

624.0

92.

2562

4.15

609.

7690

.68

1.00

949

0.65

4.70

490.

6747

3.81

92.0

41.

010

NM

30U

627.1

92.

5862

7.25

616.

6990

.62

1.02

049

7.85

7.43

497.

8447

9.94

92.2

21.

023

NM

30C

624.2

54.

8662

4.30

617.

2691

.02

1.02

149

5.70

5.36

495.

7247

8.97

92.3

41.

021

NM

40U

635.6

62.

4463

5.72

627.

0290

.94

1.03

749

9.66

6.23

499.

6848

7.33

92.7

61.

038

NM

40C

636.7

13.

6963

6.77

626.

6990

.80

1.03

750

2.63

3.22

502.

6748

5.45

92.1

41.

034

NM

50U

656.0

83.

0665

6.14

641.

0590

.48

1.06

151

6.10

0.87

516.

1549

6.19

91.6

81.

057

NM

50C

655.1

6-0

.26

655.

2363

6.45

90.8

01.

053

507.

886.

4450

7.89

495.

3292

.54

1.05

6N

M60U

673.7

01.

1067

3.77

658.

0089

.98

1.08

952

8.67

5.03

528.

7051

0.81

91.8

61.

089

NM

60C

662.1

76.

0966

2.21

652.

6691

.00

1.08

052

5.11

6.61

525.

1250

6.41

92.2

21.

079

NM

70U

697.7

7-3

.54

697.

8367

6.24

90.0

41.

119

544.

051.

9054

4.10

526.

6691

.64

1.12

2N

M70C

687.8

6-0

.04

687.

9266

6.63

90.0

21.

103

540.

659.

2954

0.62

519.

0291

.94

1.10

6N

M80U

713.7

0-1

2.34

713.

6769

7.44

89.4

21.

154

568.

36-5

.53

568.

3954

3.64

91.1

81.

158

NM

80C

695.1

6-2

.11

695.

2368

1.91

90.0

01.

128

554.

494.

4255

4.52

530.

8391

.72

1.13

1N

M90U

755.2

5-1

0.35

755.

2572

6.01

88.4

21.

201

581.

85-4

.40

581.

8956

6.27

91.1

81.

207

NM

90C

722.2

09.

5372

2.21

705.

4891

.06

1.16

756

8.22

5.07

568.

2654

4.83

91.6

81.

161

76

Tab

leE

17:

Infe

ren

cefo

rth

e84

thp

erce

nti

leeµ

+σ

bas

edon

fully

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

335.8

63.

1133

5.88

332.

4793

.74

1.00

023

3.51

4.27

233.

4923

5.72

94.9

41.

000

MI

358.7

659

.64

353.

8136

2.73

95.2

01.

091

244.

7733

.12

242.

5524

8.59

95.4

81.

055

NM

10U

336.4

33.

5333

6.45

333.

3093

.70

1.00

223

3.93

4.13

233.

9223

6.24

95.0

01.

002

NM

10C

336.5

73.

4333

6.59

333.

2393

.62

1.00

223

4.38

4.43

234.

3623

6.28

94.6

61.

002

NM

20U

339.7

44.

0033

9.75

335.

6593

.64

1.01

023

6.11

3.74

236.

1123

7.77

94.9

41.

009

NM

20C

338.4

34.

1533

8.43

335.

6293

.82

1.00

923

5.26

4.34

235.

2423

7.83

94.6

81.

009

NM

30U

342.1

71.

0034

2.20

338.

9293

.28

1.01

923

8.98

3.80

238.

9824

0.52

94.5

61.

020

NM

30C

340.9

22.

5334

0.94

338.

9393

.64

1.01

923

8.56

4.93

238.

5324

0.46

94.8

41.

020

NM

40U

349.5

12.

8034

9.53

344.

6993

.48

1.03

724

2.67

4.31

242.

6624

4.43

94.8

81.

037

NM

40C

345.8

72.

3034

5.90

343.

9093

.66

1.03

424

0.63

4.72

240.

6124

4.05

94.9

01.

035

NM

50U

355.8

24.

3435

5.83

352.

2893

.64

1.06

024

5.99

3.13

246.

0024

9.37

94.5

81.

058

NM

50C

353.1

91.

2135

3.22

349.

9293

.66

1.05

224

8.68

5.56

248.

6424

8.50

94.6

81.

054

NM

60U

363.6

45.

9536

3.62

361.

8193

.66

1.08

825

5.29

2.51

255.

3025

5.84

94.6

81.

085

NM

60C

361.4

34.

0136

1.44

357.

9893

.68

1.07

725

1.49

2.18

251.

5025

3.20

94.7

41.

074

NM

70U

378.2

3-1

.42

378.

2737

2.36

93.5

61.

120

265.

645.

9226

5.60

264.

4594

.62

1.12

2N

M70C

373.7

42.

6137

3.77

365.

9493

.46

1.10

125

7.52

3.04

257.

5325

9.15

94.6

21.

099

NM

80U

395.4

90.

0439

5.53

386.

3692

.86

1.16

227

4.67

1.44

274.

7027

3.74

94.3

41.

161

NM

80C

377.6

91.

9637

7.72

375.

1093

.30

1.12

826

4.31

2.45

264.

3326

5.40

94.4

01.

126

NM

90U

409.1

2-1

.01

409.

1640

2.08

93.2

61.

209

284.

981.

9028

5.00

285.

4094

.36

1.21

1N

M90C

386.2

55.

3938

6.25

385.

9193

.54

1.16

126

9.29

2.80

269.

3027

2.55

94.4

61.

156

77

Tab

leE

18:

Infe

ren

cefo

rth

e84

thp

erce

nti

leeµ

+σ

bas

edon

fully

mas

kedLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

104.9

82.

0910

4.97

105.

3595

.26

1.00

075

.98

1.63

75.9

774

.51

94.5

41.

000

MI

107.3

27.

5110

7.07

108.

1495

.50

1.02

677

.86

4.44

77.7

576

.23

94.4

01.

023

NM

10U

105.2

72.

1510

5.25

105.

5995

.26

1.00

276

.18

1.70

76.1

774

.67

94.3

81.

002

NM

10C

105.2

22.

2510

5.21

105.

6095

.28

1.00

275

.99

1.69

75.9

874

.67

94.5

21.

002

NM

20U

105.8

02.

0410

5.79

106.

3095

.12

1.00

976

.35

1.57

76.3

475

.17

94.7

81.

009

NM

20C

105.9

92.

2410

5.98

106.

2995

.08

1.00

976

.73

1.62

76.7

375

.16

94.6

41.

009

NM

30U

107.0

01.

9210

7.00

107.

4895

.38

1.02

077

.28

1.34

77.2

776

.01

94.3

61.

020

NM

30C

107.4

32.

8010

7.40

107.

4695

.08

1.02

077

.28

1.90

77.2

775

.98

94.4

21.

020

NM

40U

109.1

62.

0010

9.16

109.

2195

.06

1.03

778

.65

1.31

78.6

577

.23

94.3

21.

037

NM

40C

108.6

02.

2810

8.58

108.

9995

.28

1.03

578

.62

1.66

78.6

177

.07

94.4

41.

034

NM

50U

111.1

02.

8911

1.08

111.

5895

.08

1.05

979

.95

2.16

79.9

278

.89

94.5

01.

059

NM

50C

109.7

11.

7010

9.71

110.

9195

.40

1.05

380

.06

2.25

80.0

478

.49

94.5

21.

053

NM

60U

114.1

92.

0911

4.19

114.

4395

.06

1.08

682

.47

1.67

82.4

680

.93

94.0

61.

086

NM

60C

111.7

81.

5211

1.78

113.

2295

.16

1.07

581

.26

1.06

81.2

680

.06

94.2

61.

075

NM

70U

116.4

70.

5311

6.48

117.

9894

.98

1.12

084

.90

0.86

84.9

183

.47

94.5

41.

120

NM

70C

115.8

82.

7911

5.85

115.

9294

.88

1.10

083

.71

0.77

83.7

181

.90

94.6

21.

099

NM

80U

121.8

62.

4112

1.84

122.

5195

.00

1.16

387

.13

1.60

87.1

386

.62

94.7

01.

163

NM

80C

119.2

82.

6311

9.27

118.

8095

.00

1.12

885

.34

1.17

85.3

483

.96

94.5

61.

127

NM

90U

127.4

41.

4812

7.44

127.

6194

.86

1.21

191

.82

1.46

91.8

190

.26

94.4

41.

212

NM

90C

121.9

02.

2912

1.89

121.

8995

.34

1.15

787

.69

2.45

87.6

786

.21

94.3

81.

157

78

Tab

leE

19:

Infe

ren

cefo

rth

e95

thp

erce

nti

leeµ

+1.6

45σ

bas

edon

full

ym

aske

dLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103

×10

3×

103

×10

3%

Len

.

CD

1490.7

61.

2014

90.9

114

57.6

489

.12

1.00

011

64.6

12.

5411

64.7

311

26.5

291

.12

1.00

0M

I20

15.3

9691

.68

1893

.17

2268

.36

95.5

41.

556

1386

.66

380.

0113

33.7

114

40.4

895

.24

1.27

9N

M10

U14

93.4

50.

5614

93.6

014

61.4

589

.12

1.00

311

69.0

32.

0611

69.1

511

29.4

490

.98

1.00

3N

M10

C14

95.9

92.

9414

96.1

414

62.3

688

.88

1.00

311

69.5

43.

2811

69.6

511

29.7

691

.06

1.00

3N

M20

U15

00.7

2-5

.78

1500

.86

1470

.58

89.0

41.

009

1177

.00

4.13

1177

.11

1139

.26

90.9

81.

011

NM

20C

1517.9

50.

9715

18.1

014

73.4

688

.90

1.01

111

78.2

85.

6311

78.3

911

39.6

491

.22

1.01

2N

M30

U15

24.2

5-0

.01

1524

.41

1492

.78

88.9

81.

024

1200

.65

12.2

312

00.7

011

57.2

590

.94

1.02

7N

M30

C15

23.7

97.

3515

23.9

214

95.0

589

.16

1.02

611

91.1

76.

1511

91.2

711

54.1

891

.10

1.02

5N

M40

U15

54.8

01.

6515

54.9

515

23.3

088

.90

1.04

512

07.9

48.

8112

08.0

311

78.3

891

.12

1.04

6N

M40

C15

60.1

88.

4515

60.3

115

23.1

188

.78

1.04

512

12.9

50.

7912

13.0

711

72.7

490

.56

1.04

1N

M50

U16

03.2

25.

7416

03.3

715

64.6

588

.54

1.07

312

46.6

2-5

.08

1246

.74

1203

.75

90.5

81.

069

NM

50C

1604.9

6-3

.24

1605

.12

1550

.96

88.3

21.

064

1225

.10

10.4

012

25.1

812

01.2

591

.18

1.06

6N

M60

U16

73.1

45.

9816

73.3

016

17.3

188

.44

1.11

012

88.7

48.

5712

88.8

412

47.3

690

.48

1.10

7N

M60

C16

31.9

715

.78

1632

.05

1598

.46

88.7

81.

097

1277

.33

15.8

212

77.3

612

34.0

990

.90

1.09

5N

M70

U17

33.6

7-1

2.55

1733

.80

1670

.96

88.1

21.

146

1328

.07

3.66

1328

.20

1295

.26

90.2

61.

150

NM

70C

1704.6

45.

1117

04.8

016

40.4

188

.70

1.12

513

24.1

323

.39

1324

.06

1270

.89

90.6

21.

128

NM

80U

1784.1

5-3

7.01

1783

.94

1738

.89

87.1

61.

193

1407

.13

-17.

5114

07.1

613

48.2

989

.90

1.19

7N

M80

C17

20.4

7-6

.53

1720

.63

1683

.01

88.3

21.

155

1362

.23

11.6

913

62.3

213

05.1

590

.80

1.15

9N

M90

U19

01.6

4-3

0.66

1901

.59

1838

.02

86.1

41.

261

1458

.75

-12.

9314

58.8

314

23.9

389

.62

1.26

4N

M90

C18

06.2

329

.62

1806

.17

1754

.07

88.9

21.

203

1401

.73

10.8

514

01.8

313

45.3

890

.52

1.19

4

79

Tab

leE

20:

Infe

ren

cefo

rth

e95

thp

erce

nti

leeµ

+1.6

45σ

bas

edon

full

ym

aske

dLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

795.

672.

8179

5.75

795.

6693

.28

1.00

056

3.07

10.1

556

3.03

563.

8994

.16

1.00

0M

I873.

6017

7.22

855.

5290

0.05

95.7

81.

131

598.

9797

.82

590.

9960

4.98

95.5

61.

073

NM

10U

797.

463.

9079

7.53

798.

0193

.44

1.00

356

4.33

9.97

564.

3056

5.37

94.1

61.

003

NM

10C

797.

743.

1679

7.81

797.

7693

.24

1.00

356

5.35

10.7

256

5.30

565.

4794

.16

1.00

3N

M20

U805.

925.

2080

5.98

804.

6993

.34

1.01

157

0.56

8.83

570.

5556

9.68

94.2

01.

010

NM

20C

802.

575.

4880

2.63

804.

6093

.46

1.01

156

7.82

10.6

456

7.78

569.

8994

.18

1.01

1N

M30

U816.

30-2

.14

816.

3881

4.02

93.1

61.

023

579.

639.

8757

9.61

577.

5894

.14

1.02

4N

M30

C811.

741.

2681

1.82

813.

9893

.24

1.02

357

7.78

12.0

157

7.71

577.

3493

.96

1.02

4N

M40

U835.

342.

1583

5.42

830.

5093

.32

1.04

458

8.54

10.9

358

8.49

588.

6994

.26

1.04

4N

M40

C825.

831.

1982

5.91

828.

1693

.44

1.04

158

2.97

13.2

458

2.88

587.

6894

.26

1.04

2N

M50

U853.

266.

2485

3.32

852.

3393

.02

1.07

159

9.17

7.38

599.

1860

2.78

93.9

41.

069

NM

50C

847.

22-1

.17

847.

3084

5.41

93.2

01.

063

606.

4514

.85

606.

3260

0.34

94.4

41.

065

NM

60U

879.

229.

8887

9.26

880.

0593

.34

1.10

662

5.60

6.07

625.

6462

1.56

94.0

01.

102

NM

60C

870.

488.

2387

0.52

868.

8292

.86

1.09

261

2.51

6.18

612.

5461

3.71

94.1

01.

088

NM

70U

921.

83-3

.56

921.

9291

1.93

92.5

81.

146

652.

7014

.90

652.

6064

6.95

94.2

41.

147

NM

70C

908.

204.

0190

8.28

891.

7092

.84

1.12

163

2.09

7.92

632.

1063

0.73

93.7

81.

119

NM

80U

972.

78-1

.46

972.

8895

4.64

92.0

81.

200

678.

564.

3767

8.61

675.

1993

.74

1.19

7N

M80

C919.

133.

5291

9.21

918.

1392

.88

1.15

465

0.41

4.24

650.

4664

8.48

93.6

21.

150

NM

90U

101

8.3

7-4

.36

1018

.47

1006

.19

92.5

01.

265

718.

559.

3071

8.56

713.

2293

.40

1.26

5N

M90

C941.

6414

.97

941.

6294

9.73

92.9

41.

194

666.

307.

1866

6.33

669.

3293

.98

1.18

7

80

Tab

leE

21:

Infe

ren

cefo

rth

e95

thp

erce

nti

leeµ

+1.6

45σ

bas

edon

full

ym

aske

dLN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

251.2

34.

3125

1.22

251.

5995

.18

1.00

018

1.90

4.18

181.

8717

7.93

94.3

21.

000

MI

257.9

920

.96

257.

1625

9.12

95.4

41.

030

186.

9412

.61

186.

5318

2.35

94.7

01.

025

NM

10U

252.0

44.

4425

2.03

252.

2695

.18

1.00

318

2.43

4.33

182.

4017

8.40

94.3

21.

003

NM

10C

251.9

34.

7725

1.91

252.

2995

.14

1.00

318

2.04

4.28

182.

0117

8.39

94.2

41.

003

NM

20U

253.1

64.

4225

3.15

254.

2895

.10

1.01

118

2.81

4.09

182.

7817

9.81

94.5

41.

011

NM

20C

253.7

34.

6025

3.71

254.

2394

.98

1.01

018

4.03

4.29

184.

0017

9.79

94.3

61.

010

NM

30U

256.6

73.

6425

6.67

257.

6095

.26

1.02

418

5.48

3.53

185.

4718

2.18

94.6

81.

024

NM

30C

257.8

95.

8725

7.85

257.

5494

.80

1.02

418

5.40

4.90

185.

3518

2.10

94.3

01.

023

NM

40U

263.0

54.

0326

3.04

262.

5394

.86

1.04

318

9.45

3.47

189.

4418

5.64

94.3

21.

043

NM

40C

261.4

94.

6226

1.48

261.

8995

.14

1.04

118

9.25

4.08

189.

2318

5.18

94.5

21.

041

NM

50U

269.0

87.

0326

9.01

269.

3295

.20

1.07

019

3.14

5.78

193.

0719

0.38

94.4

41.

070

NM

50C

264.5

73.

1326

4.58

267.

3395

.20

1.06

319

3.29

6.25

193.

2018

9.23

94.4

81.

064

NM

60U

278.1

34.

1527

8.12

277.

5294

.72

1.10

320

0.49

4.53

200.

4619

6.28

94.3

21.

103

NM

60C

271.0

23.

1727

1.03

273.

9395

.26

1.08

919

7.27

2.88

197.

2719

3.69

94.1

61.

089

NM

70U

285.6

80.

5028

5.71

287.

9395

.04

1.14

420

7.00

2.53

207.

0120

3.72

94.5

81.

145

NM

70C

281.2

36.

6228

1.18

281.

6695

.30

1.12

020

3.38

2.41

203.

3919

8.96

94.5

21.

118

NM

80U

300.0

85.

0230

0.07

301.

5895

.06

1.19

921

5.73

3.73

215.

7221

3.18

94.6

01.

198

NM

80C

291.5

76.

4829

1.52

289.

9495

.12

1.15

220

8.02

3.34

208.

0120

4.85

94.6

21.

151

NM

90U

317.4

23.

3631

7.44

317.

8794

.72

1.26

322

8.64

4.46

228.

6222

4.84

94.5

61.

264

NM

90C

299.7

65.

6229

9.74

298.

8494

.94

1.18

821

6.01

6.32

215.

9421

1.33

94.3

61.

188

81

Tab

leE

22:

Infe

ren

cefo

rth

em

eanµ

bas

edon

full

ym

aske

dN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

181.

900.

4618

1.91

177.

4193

.56

1.00

013

9.82

4.71

139.

7613

9.02

94.4

81.

000

MI

185.

63-0

.02

185.

6518

7.70

94.8

21.

058

142.

574.

9414

2.50

144.

7695

.12

1.04

1N

M10

182.

270.

7118

2.28

177.

7993

.74

1.00

214

0.04

4.81

139.

9713

9.29

94.3

61.

002

NM

20

183.

180.

2218

3.20

178.

8993

.72

1.00

814

0.74

4.63

140.

6814

0.02

94.4

41.

007

NM

30

184.

200.

6318

4.22

180.

1693

.82

1.01

514

1.70

4.73

141.

6314

1.07

94.6

21.

015

NM

40

185.

471.

2418

5.48

181.

9594

.00

1.02

614

2.37

4.84

142.

3014

2.45

94.8

81.

025

NM

50

187.

620.

1818

7.64

183.

9693

.96

1.03

714

4.45

4.51

144.

3914

3.80

94.4

81.

034

NM

60

187.

321.

5318

7.34

185.

5093

.94

1.04

614

5.46

5.26

145.

3814

5.26

94.8

21.

045

NM

70

191.

591.

0619

1.60

188.

3094

.24

1.06

114

7.70

3.32

147.

6814

7.07

95.0

81.

058

NM

80

193.

670.

4919

3.69

190.

4294

.02

1.07

314

8.09

3.88

148.

0614

8.63

94.7

21.

069

NM

90

194.

42-0

.27

194.

4419

2.64

94.3

01.

086

151.

634.

5015

1.57

150.

4995

.22

1.08

3

82

Tab

leE

23:

Infe

ren

cefo

rth

em

eanµ

bas

edon

full

ym

aske

dN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103×

103×

103

%L

en.

CD

98.7

8-0

.57

98.7

899

.22

94.8

61.

000

69.7

7-0

.35

69.7

870

.43

95.2

61.

000

MI

101.

12-0

.94

101.

1310

2.22

94.9

01.

030

71.5

6-0

.26

71.5

772

.20

95.2

21.

025

NM

1098

.87

-0.3

898

.88

99.3

894

.72

1.00

269

.96

-0.3

369

.97

70.5

595

.26

1.00

2N

M20

99.1

5-0

.32

99.1

699

.91

95.1

21.

007

70.3

2-0

.30

70.3

370

.90

95.4

21.

007

NM

30100.

33-0

.49

100.

3410

0.60

94.8

61.

014

70.8

8-0

.36

70.8

971

.42

95.4

01.

014

NM

40101.

08-0

.40

101.

0910

1.48

94.8

01.

023

71.2

4-0

.20

71.2

571

.98

95.2

61.

022

NM

50101.

94-0

.74

101.

9510

2.43

95.0

01.

032

72.0

3-0

.32

72.0

472

.68

95.3

01.

032

NM

60103.

17-0

.62

103.

1810

3.68

94.7

41.

045

72.5

5-0

.13

72.5

673

.48

95.4

21.

043

NM

70104.

45-1

.64

104.

4410

4.90

95.0

21.

057

74.0

9-0

.21

74.1

074

.40

94.9

81.

056

NM

80105.

86-0

.70

105.

8710

6.10

95.0

01.

069

74.1

10.

0574

.12

75.2

095

.32

1.06

8N

M90

106.

67-1

.21

106.

6710

7.35

95.2

61.

082

75.1

7-0

.48

75.1

876

.21

95.0

01.

082

83

Tab

leE

24:

Infe

ren

cefo

rth

em

eanµ

bas

edon

full

ym

aske

dN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

31.

440.

5531

.44

31.5

994

.78

1.00

021

.67

-0.2

021

.67

22.3

595

.58

1.00

0M

I32.

140.

6532

.13

32.2

594

.92

1.02

122

.11

-0.1

422

.11

22.8

195

.38

1.02

0N

M10

31.

490.

5231

.49

31.6

594

.80

1.00

221

.68

-0.2

021

.68

22.3

995

.56

1.00

2N

M20

31.

620.

5131

.62

31.7

994

.94

1.00

621

.80

-0.1

921

.80

22.4

995

.46

1.00

6N

M30

31.

830.

5831

.83

32.0

194

.90

1.01

321

.96

-0.2

121

.96

22.6

495

.70

1.01

3N

M40

32.

250.

5132

.25

32.2

994

.62

1.02

222

.10

-0.2

922

.10

22.8

495

.90

1.02

2N

M50

32.

470.

5832

.47

32.6

094

.62

1.03

222

.27

-0.1

922

.27

23.0

595

.40

1.03

1N

M60

32.

830.

3632

.83

32.9

394

.64

1.04

222

.66

-0.2

322

.66

23.3

195

.58

1.04

3N

M70

33.

270.

4133

.27

33.3

194

.62

1.05

422

.84

-0.1

122

.84

23.5

795

.66

1.05

4N

M80

33.

570.

4633

.57

33.7

295

.02

1.06

723

.22

-0.1

423

.22

23.8

595

.36

1.06

7N

M90

34.

020.

3234

.02

34.1

894

.76

1.08

223

.64

-0.2

423

.64

24.1

695

.16

1.08

1

84

Tab

leE

25:

Infe

ren

cefo

rth

eva

rian

ceσ

2b

ased

onfu

lly

mas

kedN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

258.

83-3

8.93

255.

9124

8.15

88.8

81.

000

194.

38-2

4.12

192.

9019

5.18

91.8

41.

000

MI

284.

6533

.87

282.

6629

1.93

93.7

21.

176

207.

0617

.15

206.

3721

6.01

94.9

01.

107

NM

10261.

82-3

7.77

259.

1125

0.75

89.0

21.

011

195.

49-2

3.53

194.

0819

7.14

91.9

41.

010

NM

20269.

30-3

4.30

267.

1425

8.11

89.4

01.

040

201.

86-2

1.91

200.

6920

2.54

91.7

21.

038

NM

30278.

33-3

3.53

276.

3326

7.77

89.0

21.

079

208.

21-2

0.45

207.

2321

0.23

91.7

61.

077

NM

40287.

98-3

0.60

286.

3828

0.02

89.6

81.

128

215.

45-1

7.92

214.

7221

9.61

91.9

61.

125

NM

50302.

08-2

7.72

300.

8429

3.58

89.4

21.

183

229.

13-1

7.83

228.

4622

9.41

92.0

21.

175

NM

60315.

28-3

1.64

313.

7230

5.79

89.0

01.

232

234.

86-1

8.84

234.

1323

9.54

92.1

41.

227

NM

70336.

49-2

3.37

335.

7132

2.51

88.9

21.

300

248.

53-1

6.26

248.

0225

1.01

90.9

61.

286

NM

80342.

53-2

5.39

341.

6333

6.18

88.1

01.

355

262.

63-1

8.33

262.

0226

1.57

90.6

41.

340

NM

90351.

31-2

8.46

350.

1934

9.56

88.4

81.

409

270.

38-1

9.92

269.

6727

2.56

91.4

01.

396

85

Tab

leE

26:

Infe

ren

cefo

rth

eva

rian

ceσ

2b

ased

onfu

lly

mas

kedN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

142.5

8-1

0.45

142.

2113

9.94

92.8

21.

000

101.

19-5

.35

101.

0699

.47

94.1

41.

000

MI

149.2

09.

7914

8.89

148.

6194

.38

1.06

210

4.86

5.12

104.

7410

3.52

94.5

61.

041

NM

1014

4.1

9-1

0.52

143.

8214

1.27

93.0

81.

009

102.

25-5

.30

102.

1310

0.42

93.8

61.

010

NM

2014

8.1

9-8

.79

147.

9514

5.16

93.1

21.

037

105.

29-4

.32

105.

2210

3.11

93.9

01.

037

NM

3015

3.1

3-8

.51

152.

9115

0.43

93.0

01.

075

108.

97-3

.32

108.

9410

6.92

94.0

01.

075

NM

4015

9.4

5-8

.03

159.

2615

6.75

93.2

61.

120

111.

80-4

.63

111.

7211

1.21

93.9

81.

118

NM

5016

6.6

9-8

.58

166.

4816

3.59

93.0

21.

169

117.

31-4

.40

117.

2411

6.12

94.3

01.

167

NM

6017

5.6

8-5

.11

175.

6217

1.54

92.7

21.

226

123.

15-3

.59

123.

1112

1.41

93.5

21.

221

NM

7018

1.2

5-4

.39

181.

2217

9.37

93.3

01.

282

128.

33-1

.19

128.

3412

7.11

94.3

21.

278

NM

8018

9.0

5-5

.47

188.

9918

7.09

92.6

21.

337

131.

68-3

.81

131.

6413

2.37

94.3

61.

331

NM

9019

6.3

2-7

.99

196.

1819

4.73

92.6

01.

391

139.

57-3

.26

139.

5413

8.25

93.6

01.

390

86

Tab

leE

27:

Infe

ren

cefo

rth

eva

rian

ceσ

2b

ased

onfu

lly

mas

kedN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

44.

56-1

.36

44.5

444

.66

94.9

81.

000

30.7

8-0

.72

30.7

831

.60

95.7

21.

000

MI

45.

570.

6745

.57

45.7

395

.04

1.02

431

.45

0.31

31.4

532

.29

95.9

21.

022

NM

10

44.

95-1

.25

44.9

445

.10

94.7

21.

010

31.0

7-0

.80

31.0

631

.90

95.6

61.

010

NM

20

46.

24-1

.33

46.2

246

.25

95.0

81.

036

32.0

6-0

.59

32.0

632

.73

95.6

41.

036

NM

30

47.

67-1

.32

47.6

647

.91

95.1

41.

073

33.2

8-0

.83

33.2

833

.89

95.5

81.

073

NM

40

49.

71-0

.99

49.7

149

.90

95.1

21.

117

34.9

2-0

.55

34.9

235

.30

95.7

21.

117

NM

50

51.

74-0

.98

51.7

452

.09

94.9

61.

166

36.3

0-0

.99

36.2

936

.83

94.7

21.

165

NM

60

54.

06-1

.67

54.0

454

.37

94.5

81.

217

37.4

2-0

.22

37.4

238

.50

95.7

41.

218

NM

70

56.

55-1

.53

56.5

356

.79

94.9

01.

272

39.3

2-0

.64

39.3

140

.19

95.2

81.

272

NM

80

59.

01-0

.98

59.0

059

.33

95.0

81.

328

41.4

5-0

.77

41.4

541

.95

94.8

61.

328

NM

90

61.

420.

2361

.43

61.9

995

.00

1.38

842

.73

-0.6

342

.73

43.7

995

.38

1.38

6

87

Tab

leE

28:

Infe

ren

cefo

rth

e84

thp

erce

nti

leµ

+σ

bas

edon

fully

mas

kedN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

223.

68-2

7.81

221.

9621

7.29

92.5

61.

000

171.

38-1

2.26

170.

9617

0.27

94.3

01.

000

MI

227.

99-1

0.12

227.

7922

9.90

93.8

41.

058

175.

17-1

.86

175.

1817

7.30

94.9

61.

041

NM

10224.

98-2

7.14

223.

3621

8.30

92.7

21.

005

172.

05-1

1.92

171.

6517

1.04

94.3

21.

005

NM

20227.

38-2

6.34

225.

8822

1.19

92.8

21.

018

174.

35-1

1.59

173.

9817

3.17

94.2

61.

017

NM

30231.

59-2

6.14

230.

1422

5.02

92.4

21.

036

177.

59-1

1.09

177.

2617

6.26

94.2

61.

035

NM

40236.

06-2

4.73

234.

7923

0.01

92.7

01.

059

179.

07-1

0.08

178.

8118

0.12

94.5

21.

058

NM

50238.

18-2

5.28

236.

8623

5.50

92.9

61.

084

187.

43-1

1.08

187.

1218

4.17

93.9

61.

082

NM

60245.

40-2

6.95

243.

9424

0.70

92.8

61.

108

187.

93-1

1.19

187.

6218

8.55

94.4

41.

107

NM

70254.

27-2

4.78

253.

0924

7.65

92.3

61.

140

195.

12-1

2.70

194.

7319

3.38

93.7

21.

136

NM

80260.

95-2

7.02

259.

5725

3.66

92.6

61.

167

199.

83-1

4.02

199.

3619

8.06

93.4

01.

163

NM

90260.

37-3

0.12

258.

6525

9.79

92.4

01.

196

204.

79-1

4.76

204.

2820

3.15

93.4

21.

193

88

Tab

leE

29:

Infe

ren

cefo

rth

e84

thp

erce

nti

leµ

+σ

bas

edon

fully

mas

kedN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

123.3

3-8

.36

123.

0612

1.52

94.0

01.

000

86.7

2-4

.31

86.6

286

.26

94.5

61.

000

MI

126.2

9-3

.80

126.

2512

5.20

94.2

21.

030

88.9

5-1

.53

88.9

488

.43

94.4

41.

025

NM

1012

4.0

1-8

.27

123.

7412

2.03

93.8

61.

004

87.1

7-4

.30

87.0

886

.63

94.5

81.

004

NM

2012

4.9

6-7

.48

124.

7512

3.56

94.2

61.

017

88.0

2-3

.85

87.9

487

.69

94.5

01.

017

NM

3012

7.6

4-7

.69

127.

4212

5.67

94.1

21.

034

89.8

3-3

.51

89.7

789

.22

94.5

81.

034

NM

4012

9.9

8-7

.60

129.

7712

8.25

94.0

21.

055

90.7

9-4

.08

90.7

190

.98

94.6

61.

055

NM

5013

3.8

6-8

.50

133.

6013

1.10

93.9

41.

079

93.2

1-4

.24

93.1

293

.04

94.4

41.

079

NM

6013

7.7

3-7

.02

137.

5713

4.46

93.9

81.

107

95.1

9-3

.83

95.1

295

.30

94.2

41.

105

NM

7013

9.6

8-7

.93

139.

4713

7.88

94.0

41.

135

98.1

0-2

.85

98.0

797

.78

94.5

61.

134

NM

8014

3.8

1-7

.87

143.

6114

1.33

93.9

41.

163

98.9

4-4

.02

98.8

710

0.16

94.9

01.

161

NM

9014

6.1

0-1

0.03

145.

7814

4.82

94.0

61.

192

102.

43-4

.55

102.

3410

2.84

94.6

01.

192

89

Tab

leE

30:

Infe

ren

cefo

rth

e84

thp

erce

nti

leµ

+σ

bas

edon

fully

mas

kedN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

38.

35-0

.38

38.3

638

.69

95.0

21.

000

26.6

5-0

.68

26.6

527

.37

95.6

01.

000

MI

39.

240.

2339

.24

39.5

095

.08

1.02

127

.13

-0.3

527

.13

27.9

395

.82

1.02

0N

M10

38.

53-0

.35

38.5

338

.86

95.2

01.

004

26.7

3-0

.72

26.7

227

.49

95.5

41.

004

NM

20

38.

95-0

.42

38.9

539

.32

95.0

61.

016

27.1

0-0

.62

27.1

027

.82

95.6

81.

016

NM

30

39.

80-0

.37

39.8

139

.99

94.8

21.

033

27.6

9-0

.77

27.6

828

.28

95.4

41.

033

NM

40

40.

55-0

.30

40.5

640

.81

94.7

01.

055

28.1

2-0

.72

28.1

128

.86

95.8

41.

054

NM

50

41.

26-0

.24

41.2

641

.73

95.0

61.

078

28.7

1-0

.85

28.7

029

.51

95.7

81.

078

NM

60

42.

33-0

.84

42.3

242

.71

95.0

61.

104

29.4

1-0

.51

29.4

130

.23

95.4

61.

104

NM

70

43.

28-0

.76

43.2

843

.78

95.3

61.

131

30.2

1-0

.63

30.2

030

.97

95.6

61.

131

NM

80

44.

72-0

.47

44.7

344

.91

95.0

21.

161

31.1

3-0

.74

31.1

331

.76

95.1

41.

160

NM

90

45.

25-0

.04

45.2

646

.13

95.5

21.

192

31.7

7-0

.78

31.7

732

.61

95.1

81.

191

90

Tab

leE

31:

Infe

ren

cefo

rth

e95

thp

erce

nti

leµ

+1.6

45σ

bas

edon

full

ym

aske

dN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

282.

20-4

6.04

278.

4527

2.14

92.2

01.

000

214.

79-2

3.21

213.

5521

3.25

94.2

41.

000

MI

287.

75-1

6.64

287.

3028

7.96

93.6

01.

058

219.

67-6

.24

219.

6022

2.07

95.0

01.

041

NM

10284.

35-4

5.10

280.

7827

3.92

91.9

81.

007

215.

83-2

2.71

214.

6621

4.62

93.9

61.

006

NM

20288.

75-4

3.47

285.

4827

8.92

92.1

81.

025

220.

02-2

2.05

218.

9421

8.38

93.9

41.

024

NM

30295.

93-4

3.40

292.

7628

5.74

91.9

81.

050

225.

38-2

1.29

224.

4022

3.83

93.8

41.

050

NM

40303.

53-4

1.48

300.

7129

4.40

92.0

21.

082

228.

73-1

9.70

227.

9023

0.54

94.2

01.

081

NM

50308.

60-4

1.71

305.

8030

3.94

92.5

81.

117

241.

86-2

1.13

240.

9623

7.66

93.5

61.

114

NM

60321.

40-4

5.32

318.

2231

3.18

92.1

61.

151

243.

67-2

1.81

242.

7124

5.29

94.0

21.

150

NM

70335.

73-4

1.44

333.

1932

4.76

91.7

01.

193

255.

54-2

3.03

254.

5325

3.55

93.1

21.

189

NM

80345.

34-4

4.77

342.

4633

5.11

91.7

81.

231

264.

74-2

5.56

263.

5326

1.63

93.0

21.

227

NM

90346.

83-4

9.38

343.

3334

5.56

92.1

81.

270

271.

85-2

7.19

270.

5227

0.22

93.0

81.

267

91

Tab

leE

32:

Infe

ren

cefo

rth

e95

thp

erce

nti

leµ

+1.6

45σ

bas

edon

full

ym

aske

dN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

155.5

0-1

3.39

154.

9315

2.20

93.8

01.

000

109.

38-6

.87

109.

1710

8.04

94.5

61.

000

MI

159.2

9-5

.64

159.

2015

6.82

94.2

21.

030

112.

18-2

.36

112.

1711

0.76

94.6

41.

025

NM

1015

6.7

3-1

3.35

156.

1815

3.13

93.9

21.

006

110.

16-6

.86

109.

9610

8.71

94.4

21.

006

NM

2015

8.7

6-1

2.09

158.

3115

5.83

94.2

41.

024

111.

82-6

.14

111.

6611

0.59

94.6

21.

024

NM

3016

3.0

0-1

2.33

162.

5415

9.59

94.0

21.

049

114.

85-5

.54

114.

7211

3.30

94.2

81.

049

NM

4016

7.2

0-1

2.24

166.

7716

4.14

93.7

61.

078

116.

70-6

.59

116.

5311

6.45

94.6

21.

078

NM

5017

3.6

1-1

3.51

173.

1016

9.17

93.4

81.

111

120.

87-6

.78

120.

6912

0.05

94.6

01.

111

NM

6018

0.3

0-1

1.15

179.

9717

4.91

93.5

61.

149

124.

69-6

.21

124.

5512

3.96

94.5

61.

147

NM

7018

3.7

4-1

2.00

183.

3718

0.77

93.3

41.

188

128.

97-4

.56

128.

9012

8.18

94.5

21.

186

NM

8019

0.2

7-1

2.50

189.

8818

6.68

93.6

61.

227

130.

98-6

.65

130.

8313

2.29

94.7

41.

225

NM

9019

4.9

8-1

5.71

194.

3619

2.64

93.9

61.

266

137.

10-7

.17

136.

9313

6.79

94.6

21.

266

92

Tab

leE

33:

Infe

ren

cefo

rth

e95

thp

erce

nti

leµ

+1.6

45σ

bas

edon

full

ym

aske

dN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

48.

05-0

.98

48.0

548

.46

95.0

01.

000

33.4

3-0

.98

33.4

234

.28

95.5

41.

000

MI

49.

15-0

.03

49.1

649

.47

95.0

21.

021

34.0

3-0

.49

34.0

334

.98

95.7

21.

020

NM

10

48.

36-0

.92

48.3

548

.77

95.2

41.

006

33.5

9-1

.05

33.5

834

.50

95.4

81.

006

NM

20

49.

17-1

.02

49.1

649

.59

94.8

61.

023

34.2

8-0

.89

34.2

735

.08

95.5

61.

023

NM

30

50.

56-0

.97

50.5

550

.78

95.2

01.

048

35.2

8-1

.12

35.2

635

.92

95.2

61.

048

NM

40

51.

87-0

.82

51.8

652

.23

95.1

41.

078

36.1

9-1

.00

36.1

836

.94

95.7

01.

078

NM

50

53.

21-0

.77

53.2

153

.84

95.0

81.

111

37.2

3-1

.28

37.2

138

.07

95.7

01.

111

NM

60

55.

03-1

.61

55.0

155

.56

94.7

41.

146

38.2

5-0

.70

38.2

539

.32

95.3

81.

147

NM

70

56.

72-1

.51

56.7

057

.39

95.3

81.

184

39.6

9-0

.96

39.6

840

.60

95.7

41.

184

NM

80

59.

06-1

.06

59.0

659

.32

94.9

81.

224

41.2

6-1

.13

41.2

541

.95

95.0

01.

224

NM

90

60.

19-0

.27

60.1

961

.36

95.4

21.

266

42.2

5-1

.13

42.2

443

.37

95.7

61.

265

93

Tab

leE

34:

Infe

ren

cefo

rµ

+σ

2/2

bas

edon

full

ym

aske

dN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=30

Res

ult

sfo

rn

=50

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

221.

49-1

9.01

220.

7021

6.90

92.6

61.

000

170.

31-7

.35

170.

1717

0.03

94.3

81.

000

MI

231.

7116

.91

231.

1223

8.29

94.6

21.

099

176.

8313

.51

176.

3318

0.82

95.5

01.

063

NM

10

222.

92-1

8.18

222.

2021

7.98

92.5

81.

005

171.

02-6

.95

170.

8917

0.83

94.4

41.

005

NM

20

225.

68-1

6.93

225.

0622

1.04

92.7

81.

019

173.

38-6

.32

173.

2817

3.02

94.4

01.

018

NM

30

229.

87-1

6.13

229.

3322

4.97

92.3

61.

037

176.

69-5

.50

176.

6317

6.18

94.4

41.

036

NM

40

234.

40-1

4.06

234.

0023

0.16

92.6

21.

061

178.

28-4

.12

178.

2518

0.16

94.5

81.

060

NM

50

236.

99-1

3.68

236.

6223

5.92

93.0

21.

088

186.

90-4

.40

186.

8618

4.30

93.8

81.

084

NM

60

244.

35-1

4.30

243.

9624

1.12

92.7

81.

112

187.

40-4

.16

187.

3818

8.69

94.6

01.

110

NM

70

253.

99-1

0.63

253.

7924

8.76

92.4

41.

147

194.

35-4

.81

194.

3119

3.74

93.7

81.

139

NM

80

260.

01-1

2.21

259.

7525

4.84

92.6

01.

175

199.

37-5

.28

199.

3219

8.46

93.5

61.

167

NM

90

259.

20-1

4.50

258.

8226

0.97

92.3

21.

203

204.

13-5

.46

204.

0820

3.56

93.3

81.

197

94

Tab

leE

35:

Infe

ren

cefo

rµ

+σ

2/2

bas

edon

full

ym

aske

dN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

123.

16-5

.79

123.

0312

1.48

94.0

01.

000

86.6

0-3

.02

86.5

586

.24

94.5

41.

000

MI

126.

953.

9612

6.90

126.

4594

.50

1.04

189

.15

2.30

89.1

388

.86

94.8

01.

030

NM

10123.

85-5

.64

123.

7412

1.99

93.8

41.

004

87.0

7-2

.98

87.0

386

.62

94.6

01.

004

NM

20124.

88-4

.71

124.

8112

3.57

94.2

81.

017

87.9

5-2

.46

87.9

387

.69

94.5

61.

017

NM

30127.

53-4

.74

127.

4512

5.69

94.1

21.

035

89.7

7-2

.02

89.7

689

.24

94.5

21.

035

NM

40129.

97-4

.42

129.

9112

8.30

94.1

41.

056

90.6

5-2

.51

90.6

290

.99

94.6

01.

055

NM

50133.

90-5

.03

133.

8213

1.17

93.9

21.

080

93.0

9-2

.52

93.0

793

.06

94.4

41.

079

NM

60137.

79-3

.17

137.

7713

4.66

93.8

01.

109

95.1

1-1

.93

95.1

095

.35

94.3

01.

106

NM

70139.

71-3

.83

139.

6713

8.13

93.8

41.

137

98.2

3-0

.81

98.2

497

.89

94.5

01.

135

NM

80144.

12-3

.44

144.

1014

1.59

93.8

41.

166

98.9

3-1

.85

98.9

210

0.23

94.9

21.

162

NM

90146.

08-5

.20

146.

0114

5.06

94.0

41.

194

102.

28-2

.11

102.

2710

2.94

94.7

21.

194

95

Tab

leE

36:

Infe

ren

cefo

rµ

+σ

2/2

bas

edon

full

ym

aske

dN

(µ=

0,σ

2=

1)d

ata

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

38.

36-0

.13

38.3

638

.69

95.0

61.

000

26.6

4-0

.56

26.6

427

.37

95.6

21.

000

MI

39.

280.

9839

.27

39.5

395

.14

1.02

227

.13

0.01

27.1

327

.94

95.8

21.

021

NM

10

38.

53-0

.10

38.5

438

.86

95.2

61.

004

26.7

2-0

.60

26.7

127

.49

95.5

01.

004

NM

20

38.

96-0

.15

38.9

639

.32

95.0

81.

016

27.0

9-0

.49

27.0

927

.82

95.6

41.

016

NM

30

39.

80-0

.08

39.8

039

.98

94.7

81.

033

27.6

8-0

.63

27.6

728

.28

95.4

21.

033

NM

40

40.

560.

0140

.56

40.8

194

.74

1.05

528

.11

-0.5

728

.11

28.8

695

.84

1.05

5N

M50

41.

260.

0941

.26

41.7

395

.10

1.07

928

.70

-0.6

928

.70

29.5

195

.78

1.07

8N

M60

42.

32-0

.47

42.3

242

.71

95.1

21.

104

29.4

1-0

.34

29.4

130

.23

95.4

61.

104

NM

70

43.

26-0

.36

43.2

743

.78

95.3

21.

131

30.2

0-0

.43

30.2

030

.97

95.6

41.

132

NM

80

44.

73-0

.03

44.7

344

.92

94.9

41.

161

31.1

2-0

.52

31.1

231

.76

95.1

61.

160

NM

90

45.

260.

4445

.26

46.1

595

.44

1.19

331

.76

-0.5

631

.76

32.6

195

.18

1.19

1

96

Tab

leE

37:

Infe

ren

cefo

rth

em

ean

bas

edon

Exp

onen

tial

(mea

n=

1)d

ata

wit

hC

=2.

996

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103×

103×

103

%L

en.

CD

99.

45-0

.65

99.4

699

.93

94.6

41.

000

70.7

1-1

.16

70.7

170

.63

94.8

61.

000

MI.

C90

104.3

6-4

.31

104.

2999

.75

93.3

20.

998

73.9

8-3

.12

73.9

270

.59

93.4

80.

999

MI.

C80

108.7

50.

9510

8.76

100.

5092

.78

1.00

676

.28

-0.6

476

.28

70.8

993

.08

1.00

4M

I.D

9076

9.4

213

7.33

757.

1416

1.17

95.2

01.

613

101.

3537

.12

94.3

276

.72

95.2

41.

086

MI.

D80

117.6

260

.71

100.

7511

1.41

96.1

41.

115

76.8

326

.35

72.1

875

.19

95.7

01.

065

CE

NS

102.4

70.

6710

2.47

102.

5595

.36

1.02

672

.57

-0.2

272

.57

72.5

295

.04

1.02

7N

M10

.199.

54-0

.50

99.5

410

0.07

94.6

81.

001

70.8

8-1

.06

70.8

770

.72

94.6

21.

001

NM

10.2

99.

56-0

.50

99.5

710

0.08

94.6

81.

001

70.8

8-1

.08

70.8

770

.73

94.6

41.

001

NM

20.1

99.

94-0

.59

99.9

510

0.36

94.6

21.

004

71.0

7-1

.00

71.0

770

.94

94.7

21.

004

NM

20.2

99.

99-0

.54

100.

0010

0.42

94.7

01.

005

71.0

9-1

.06

71.0

970

.98

94.6

61.

005

NM

30.1

100.1

2-0

.29

100.

1310

0.75

94.6

41.

008

71.2

7-0

.85

71.2

771

.20

94.8

01.

008

NM

30.2

100.3

1-0

.10

100.

3210

0.96

94.8

01.

010

71.4

0-0

.92

71.4

071

.33

94.7

61.

010

NM

40.1

100.4

1-0

.66

100.

4210

1.02

94.6

61.

011

71.4

5-0

.72

71.4

571

.44

94.8

21.

011

NM

40.2

100.6

7-0

.77

100.

6810

1.48

94.7

61.

015

71.6

2-0

.78

71.6

271

.77

94.8

81.

016

NM

50.1

100.9

00.

1210

0.91

101.

3694

.82

1.01

471

.61

-0.6

871

.61

71.6

194

.90

1.01

4N

M50

.210

1.7

40.

5410

1.74

102.

3594

.76

1.02

472

.22

-0.8

072

.22

72.2

794

.58

1.02

3N

M60

.110

0.9

2-0

.14

100.

9310

1.50

94.5

81.

016

71.7

3-0

.74

71.7

371

.72

95.1

01.

015

NM

60.2

102.2

40.

2210

2.25

103.

1994

.66

1.03

372

.60

-0.9

272

.60

72.8

794

.86

1.03

2N

M70

.110

1.0

80.

0310

1.09

101.

6594

.68

1.01

771

.89

-0.4

671

.89

71.8

394

.68

1.01

7N

M70

.210

3.8

20.

7210

3.83

104.

3294

.68

1.04

473

.55

-0.2

873

.56

73.6

794

.94

1.04

3N

M80

.110

1.4

70.

2110

1.48

101.

7694

.74

1.01

871

.88

-0.4

771

.89

71.9

095

.10

1.01

8N

M80

.210

5.4

91.

0210

5.50

105.

6494

.80

1.05

774

.64

-0.2

074

.65

74.5

894

.80

1.05

6N

M90

.110

1.4

00.

1410

1.41

101.

8394

.62

1.01

971

.83

-0.5

571

.84

71.9

494

.88

1.01

9N

M90

.210

7.0

91.

2110

7.09

107.

1794

.68

1.07

275

.43

-0.6

575

.43

75.5

994

.70

1.07

0

97

Tab

leE

38:

Infe

ren

cefo

rth

em

ean

bas

edon

Exp

onen

tial

(mea

n=

1)d

ata

wit

hC

=2.

996

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

31.9

5-0

.02

31.9

631

.62

94.8

61.

000

22.3

80.

4522

.37

22.3

794

.82

1.00

0M

I.C

90

33.5

1-0

.34

33.5

131

.65

93.7

01.

001

23.4

40.

3523

.44

22.3

993

.84

1.00

1M

I.C

80

34.5

90.

4134

.59

31.7

292

.84

1.00

324

.18

0.78

24.1

722

.43

93.0

01.

003

MI.

D90

33.3

06.

7232

.62

32.5

895

.04

1.03

023

.25

3.79

22.9

422

.95

94.7

01.

026

MI.

D80

33.2

35.

2932

.81

32.7

094

.88

1.03

423

.14

3.07

22.9

423

.05

94.8

81.

030

CE

NS

32.8

70.

0532

.87

32.4

494

.54

1.02

622

.96

0.63

22.9

622

.95

95.0

41.

026

NM

10.

132

.04

-0.0

232

.04

31.6

694

.76

1.00

122

.41

0.48

22.4

122

.40

95.0

21.

001

NM

10.

232

.06

-0.0

132

.06

31.6

694

.70

1.00

122

.41

0.48

22.4

122

.40

94.9

61.

001

NM

20.

132

.07

-0.0

332

.07

31.7

694

.88

1.00

422

.45

0.46

22.4

522

.47

94.9

01.

004

NM

20.

232

.07

-0.0

232

.07

31.7

794

.80

1.00

522

.46

0.44

22.4

522

.48

94.8

21.

005

NM

30.

132

.25

-0.0

832

.25

31.8

794

.80

1.00

822

.55

0.51

22.5

522

.55

95.0

81.

008

NM

30.

232

.30

-0.0

532

.30

31.9

394

.96

1.01

022

.59

0.54

22.5

922

.59

94.9

81.

010

NM

40.

132

.30

0.04

32.3

031

.97

94.9

21.

011

22.6

40.

5522

.64

22.6

294

.90

1.01

1N

M40.

232

.38

0.08

32.3

932

.12

94.9

01.

016

22.7

70.

5522

.76

22.7

294

.84

1.01

6N

M50.

132

.38

-0.0

432

.39

32.0

494

.68

1.01

322

.73

0.54

22.7

322

.67

94.9

41.

013

NM

50.

232

.72

0.08

32.7

232

.35

94.5

01.

023

23.0

20.

5823

.02

22.8

894

.54

1.02

3N

M60.

132

.52

0.01

32.5

232

.09

95.0

21.

015

22.7

40.

5122

.74

22.7

195

.04

1.01

5N

M60.

233

.11

0.04

33.1

132

.62

94.6

81.

031

23.0

90.

5023

.08

23.0

895

.18

1.03

1N

M70.

132

.46

0.02

32.4

632

.13

94.7

01.

016

22.7

80.

5122

.78

22.7

394

.96

1.01

6N

M70.

233

.26

0.18

33.2

632

.96

94.9

21.

042

23.5

20.

5123

.52

23.3

194

.74

1.04

2N

M80.

132

.56

-0.0

532

.57

32.1

694

.48

1.01

722

.74

0.61

22.7

422

.75

94.9

61.

017

NM

80.

233

.69

0.20

33.6

933

.36

94.6

21.

055

23.5

80.

7323

.57

23.6

095

.12

1.05

5N

M90.

132

.63

-0.0

132

.64

32.1

894

.64

1.01

822

.73

0.56

22.7

322

.77

95.0

41.

018

NM

90.

234

.29

0.19

34.2

933

.82

94.4

01.

070

23.8

50.

5323

.84

23.9

294

.84

1.06

9

98

Tab

leE

39:

Infe

ren

cefo

rth

em

ean

bas

edon

Exp

onen

tial

(mea

n=

1)d

ata

wit

hC

=2.

303

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103×

103×

103

%L

en.

CD

98.3

50.

6298

.35

100.

0694

.66

1.00

070

.32

-0.1

570

.32

70.7

094

.68

1.00

0M

I.C

90

107.

681.

9510

7.67

100.

5993

.02

1.00

577

.00

0.86

77.0

070

.99

92.3

61.

004

MI.

C80

114.

3713

.20

113.

6210

2.25

92.8

21.

022

80.0

35.

2879

.86

71.6

091

.84

1.01

3M

I.D

90

121.

1159

.58

105.

4611

1.53

96.2

61.

115

77.5

927

.31

72.6

375

.28

95.6

21.

065

MI.

D80

109.

2241

.89

100.

8810

8.86

97.0

21.

088

74.7

319

.96

72.0

374

.78

95.8

01.

058

CE

NS

104.

653.

9910

4.59

105.

6195

.44

1.05

574

.66

1.57

74.6

674

.59

95.0

21.

055

NM

10.

198

.59

0.71

98.6

010

0.25

94.7

21.

002

70.4

3-0

.05

70.4

470

.84

94.7

61.

002

NM

10.

298

.59

0.69

98.6

010

0.26

94.7

01.

002

70.4

4-0

.06

70.4

570

.84

94.7

41.

002

NM

20.

198

.95

0.85

98.9

610

0.73

94.8

01.

007

70.6

3-0

.10

70.6

371

.16

94.7

01.

007

NM

20.

298

.93

0.78

98.9

310

0.78

94.8

61.

007

70.7

0-0

.09

70.7

171

.20

94.8

01.

007

NM

30.

199

.98

1.65

99.9

810

1.42

94.8

61.

014

71.2

2-0

.02

71.2

271

.60

94.6

41.

013

NM

30.

2100.

151.

6310

0.14

101.

6194

.70

1.01

571

.37

0.01

71.3

871

.74

94.7

81.

015

NM

40.

1100.

511.

4310

0.51

102.

0194

.94

1.01

971

.63

0.29

71.6

472

.05

94.7

21.

019

NM

40.

2101.

001.

4110

1.00

102.

5095

.00

1.02

471

.94

0.32

71.9

572

.40

94.9

01.

024

NM

50.

1100.

681.

7510

0.68

102.

5695

.16

1.02

572

.04

0.45

72.0

472

.42

94.7

21.

024

NM

50.

2101.

571.

6310

1.57

103.

6095

.02

1.03

572

.86

0.44

72.8

773

.17

94.7

61.

035

NM

60.

1101.

523.

0010

1.48

103.

1095

.02

1.03

072

.85

0.67

72.8

572

.72

94.3

41.

029

NM

60.

2103.

563.

1210

3.53

105.

0495

.24

1.05

074

.58

0.72

74.5

874

.08

94.3

81.

048

NM

70.

1101.

882.

4810

1.86

103.

3594

.88

1.03

372

.90

0.78

72.9

072

.93

94.7

41.

032

NM

70.

2105.

653.

2610

5.61

106.

6395

.00

1.06

675

.26

0.92

75.2

675

.20

94.7

81.

064

NM

80.

1102.

002.

5610

1.98

103.

5895

.06

1.03

573

.10

0.96

73.1

073

.10

94.5

81.

034

NM

80.

2106.

882.

6710

6.85

108.

4694

.96

1.08

476

.48

1.30

76.4

876

.56

94.7

81.

083

NM

90.

1102.

192.

7310

2.16

103.

7994

.96

1.03

773

.27

0.99

73.2

773

.23

94.4

01.

036

NM

90.

2109.

694.

7410

9.60

111.

0295

.10

1.10

978

.28

0.06

78.2

978

.06

94.5

01.

104

99

Tab

leE

40:

Infe

ren

cefo

rth

em

ean

bas

edon

Exp

onen

tial

(mea

n=

1)d

ata

wit

hC

=2.

303

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

31.1

50.

1831

.16

31.6

395

.34

1.00

022

.26

0.11

22.2

622

.36

95.0

41.

000

MI.

C90

33.8

60.

2933

.87

31.7

193

.34

1.00

324

.23

0.07

24.2

422

.42

92.8

41.

002

MI.

C80

35.1

91.

2735

.17

31.8

592

.12

1.00

725

.02

0.67

25.0

122

.50

92.2

21.

006

MI.

D90

32.4

05.

2931

.97

32.7

095

.56

1.03

422

.98

2.61

22.8

423

.04

95.3

21.

030

MI.

D80

32.5

54.

1432

.29

32.7

295

.16

1.03

423

.04

2.16

22.9

423

.07

95.2

61.

032

CE

NS

32.9

60.

3832

.97

33.3

495

.12

1.05

423

.64

0.20

23.6

423

.57

95.0

41.

054

NM

10.

131

.19

0.21

31.1

931

.69

95.2

41.

002

22.3

20.

1222

.32

22.4

095

.04

1.00

2N

M10.

231

.20

0.20

31.2

031

.69

95.2

61.

002

22.3

20.

1322

.32

22.4

195

.04

1.00

2N

M20.

131

.42

0.31

31.4

231

.84

95.3

01.

007

22.4

40.

0822

.44

22.5

195

.06

1.00

7N

M20.

231

.45

0.34

31.4

531

.86

95.3

61.

007

22.4

40.

0822

.44

22.5

295

.04

1.00

7N

M30.

131

.51

0.34

31.5

132

.03

95.0

61.

013

22.6

20.

0622

.62

22.6

495

.04

1.01

3N

M30.

231

.59

0.39

31.5

932

.10

95.1

21.

015

22.6

10.

0722

.61

22.6

995

.30

1.01

5N

M40.

131

.82

0.26

31.8

332

.22

95.1

01.

019

22.6

70.

1322

.67

22.7

895

.10

1.01

9N

M40.

232

.05

0.29

32.0

532

.38

95.0

01.

024

22.8

00.

1022

.80

22.8

995

.08

1.02

4N

M50.

132

.01

0.29

32.0

132

.38

95.0

21.

024

22.8

50.

2622

.85

22.9

095

.20

1.02

4N

M50.

232

.49

0.37

32.4

932

.72

94.8

81.

034

23.0

30.

3123

.03

23.1

394

.90

1.03

4N

M60.

132

.18

0.21

32.1

832

.50

95.2

81.

028

22.9

00.

1222

.90

22.9

894

.86

1.02

8N

M60.

232

.91

0.30

32.9

233

.11

95.5

41.

047

23.3

10.

0823

.31

23.4

195

.22

1.04

7N

M70.

132

.06

0.27

32.0

632

.59

95.2

41.

031

23.0

40.

2023

.04

23.0

494

.64

1.03

0N

M70.

232

.99

0.51

32.9

933

.61

95.2

61.

063

23.7

80.

1623

.78

23.7

695

.30

1.06

2N

M80.

132

.18

0.18

32.1

832

.66

95.3

01.

033

23.0

60.

0923

.06

23.0

995

.02

1.03

3N

M80.

233

.81

0.18

33.8

234

.19

95.1

81.

081

24.1

50.

1424

.15

24.1

895

.02

1.08

1N

M90.

132

.22

0.16

32.2

232

.71

95.2

81.

034

23.0

50.

1123

.05

23.1

395

.16

1.03

4N

M90.

234

.33

0.14

34.3

334

.91

95.2

21.

104

24.5

00.

1124

.50

24.6

895

.22

1.10

4

100

Tab

leE

41:

Infe

ren

cefo

rth

elo

gsc

ale

mea

nµ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

97.9

1-0

.60

97.9

299

.25

95.0

01.

000

70.5

3-0

.68

70.5

370

.41

95.1

81.

000

MI.

C90

98.9

9-2

.34

98.9

899

.03

94.8

20.

998

71.2

7-1

.85

71.2

570

.30

94.9

60.

998

MI.

C80

100

.33

-0.4

210

0.34

99.4

294

.48

1.00

271

.71

-0.6

271

.71

70.4

994

.78

1.00

1M

I.D

90

98.8

0-0

.78

98.8

010

2.24

95.3

41.

030

70.5

5-0

.64

70.5

570

.60

95.2

41.

003

MI.

D80

98.1

0-0

.67

98.1

099

.85

95.0

81.

006

70.6

1-0

.68

70.6

170

.63

95.1

81.

003

CE

NS

98.5

90.

1198

.60

99.8

095

.04

1.00

670

.89

-0.3

770

.89

70.7

395

.18

1.00

5N

M10.

197.9

3-0

.58

97.9

499

.26

94.9

01.

000

70.5

4-0

.70

70.5

470

.41

95.1

61.

000

NM

10.

297.9

3-0

.58

97.9

399

.26

94.9

21.

000

70.5

4-0

.70

70.5

470

.41

95.1

61.

000

NM

20.

197.9

7-0

.57

97.9

899

.28

94.9

61.

000

70.5

4-0

.66

70.5

470

.43

95.2

61.

000

NM

20.

297.9

7-0

.58

97.9

899

.29

94.9

41.

000

70.5

5-0

.67

70.5

570

.44

95.2

41.

000

NM

30.

197.9

9-0

.49

98.0

099

.32

95.0

61.

001

70.5

6-0

.60

70.5

670

.46

95.1

61.

001

NM

30.

298.0

2-0

.50

98.0

399

.34

95.0

41.

001

70.5

8-0

.61

70.5

970

.47

95.1

21.

001

NM

40.

198.0

1-0

.44

98.0

299

.36

94.8

81.

001

70.5

9-0

.66

70.5

970

.47

95.1

41.

001

NM

40.

298.0

5-0

.46

98.0

699

.40

95.0

41.

002

70.6

0-0

.73

70.6

070

.49

95.2

01.

001

NM

50.

198.1

0-0

.39

98.1

199

.40

94.9

61.

002

70.6

4-0

.61

70.6

570

.50

95.2

21.

001

NM

50.

298.1

6-0

.38

98.1

799

.49

95.0

81.

002

70.7

2-0

.62

70.7

270

.56

95.0

41.

002

NM

60.

198.2

3-0

.41

98.2

399

.43

94.9

41.

002

70.6

5-0

.50

70.6

570

.54

95.1

41.

002

NM

60.

298.5

0-0

.45

98.5

199

.60

94.9

61.

004

70.7

6-0

.57

70.7

770

.66

94.9

61.

004

NM

70.

198.2

3-0

.30

98.2

499

.48

95.0

01.

002

70.6

9-0

.63

70.6

970

.53

95.0

61.

002

NM

70.

298.5

9-0

.46

98.6

099

.78

94.8

61.

005

70.9

1-0

.84

70.9

170

.75

95.0

61.

005

NM

80.

198.1

7-0

.37

98.1

899

.48

94.9

41.

002

70.7

8-0

.49

70.7

970

.57

95.1

61.

002

NM

80.

298.6

8-0

.52

98.6

910

0.05

94.9

21.

008

71.1

7-0

.42

71.1

770

.99

94.9

61.

008

NM

90.

198.1

8-0

.25

98.1

999

.53

95.0

41.

003

70.6

9-0

.55

70.6

970

.57

95.2

41.

002

NM

90.

299.1

1-0

.34

99.1

210

0.59

94.9

81.

013

71.1

4-0

.69

71.1

571

.32

95.4

21.

013

101

Tab

leE

42:

Infe

ren

cefo

rth

elo

gsc

ale

mea

nµ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

31.5

9-0

.43

31.5

931

.59

95.2

01.

000

22.4

6-0

.07

22.4

622

.35

94.6

41.

000

MI.

C90

31.8

1-0

.69

31.8

131

.59

95.0

01.

000

22.6

3-0

.15

22.6

322

.36

94.4

81.

000

MI.

C80

32.2

2-0

.28

32.2

231

.62

94.7

61.

001

22.7

8-0

.07

22.7

822

.37

94.5

01.

001

MI.

D90

31.6

0-0

.42

31.6

031

.61

95.1

81.

001

22.4

6-0

.06

22.4

622

.37

94.6

41.

001

MI.

D80

31.6

1-0

.44

31.6

131

.63

95.1

21.

001

22.4

6-0

.05

22.4

622

.38

94.6

41.

001

CE

NS

31.6

9-0

.38

31.6

931

.71

95.1

01.

004

22.5

20.

0222

.53

22.4

494

.64

1.00

4N

M10.

131

.59

-0.4

231

.59

31.5

995

.22

1.00

022

.46

-0.0

622

.46

22.3

694

.66

1.00

0N

M10.

231

.59

-0.4

231

.59

31.5

995

.20

1.00

022

.46

-0.0

622

.46

22.3

694

.66

1.00

0N

M20.

131

.60

-0.4

231

.60

31.6

095

.34

1.00

022

.47

-0.0

622

.47

22.3

694

.54

1.00

0N

M20.

231

.60

-0.4

231

.60

31.6

095

.26

1.00

022

.47

-0.0

622

.48

22.3

694

.54

1.00

0N

M30.

131

.59

-0.4

231

.59

31.6

195

.22

1.00

122

.47

-0.0

622

.47

22.3

794

.70

1.00

1N

M30.

231

.61

-0.4

231

.61

31.6

195

.22

1.00

122

.48

-0.0

622

.49

22.3

794

.64

1.00

1N

M40.

131

.62

-0.4

031

.62

31.6

295

.30

1.00

122

.47

-0.0

622

.47

22.3

794

.68

1.00

1N

M40.

231

.66

-0.4

031

.66

31.6

395

.48

1.00

122

.49

-0.0

622

.49

22.3

894

.60

1.00

1N

M50.

131

.60

-0.4

031

.60

31.6

395

.20

1.00

122

.48

-0.0

422

.48

22.3

894

.72

1.00

1N

M50.

231

.64

-0.4

131

.64

31.6

695

.34

1.00

222

.51

-0.0

522

.51

22.4

094

.62

1.00

2N

M60.

131

.65

-0.4

231

.65

31.6

495

.14

1.00

122

.47

-0.0

322

.47

22.3

994

.66

1.00

1N

M60.

231

.72

-0.4

431

.72

31.6

995

.20

1.00

322

.52

-0.0

122

.52

22.4

394

.58

1.00

3N

M70.

131

.64

-0.4

431

.64

31.6

495

.20

1.00

222

.50

-0.0

122

.50

22.3

994

.78

1.00

2N

M70.

231

.74

-0.4

431

.74

31.7

595

.14

1.00

522

.58

-0.0

322

.58

22.4

794

.62

1.00

5N

M80.

131

.63

-0.4

131

.63

31.6

595

.38

1.00

222

.50

-0.0

422

.50

22.4

094

.78

1.00

2N

M80.

231

.78

-0.4

431

.78

31.8

495

.22

1.00

822

.68

-0.0

322

.69

22.5

394

.76

1.00

8N

M90.

131

.61

-0.4

131

.61

31.6

595

.26

1.00

222

.50

-0.0

222

.50

22.4

094

.76

1.00

2N

M90.

231

.96

-0.4

631

.96

32.0

095

.30

1.01

322

.70

-0.0

122

.70

22.6

594

.84

1.01

3

102

Tab

leE

43:

Infe

ren

cefo

rth

elo

gsc

ale

vari

anceσ

2b

ased

onLN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

139.8

9-1

0.04

139.

5414

0.00

93.6

21.

000

99.1

3-5

.98

98.9

699

.40

93.9

01.

000

MI.

C90

149.4

1-1

4.14

148.

7613

9.95

91.9

81.

000

105.

08-9

.13

104.

6999

.35

92.5

61.

000

MI.

C80

153.1

4-7

.06

152.

9914

1.48

91.7

41.

011

107.

75-4

.53

107.

6610

0.09

92.0

81.

007

MI.

D90

18835.

3134

6.71

1883

4.00

620.

6795

.06

4.43

310

1.57

-1.5

510

1.57

100.

5794

.22

1.01

2M

I.D

80

141.7

7-0

.38

141.

7814

2.67

94.0

21.

019

100.

13-1

.67

100.

1310

0.43

94.1

81.

010

CE

NS

146.2

4-6

.57

146.

1114

6.52

93.5

41.

047

103.

28-4

.51

103.

1910

3.72

93.8

61.

043

NM

10.1

140.0

0-9

.98

139.

6614

0.10

93.4

81.

001

99.2

4-6

.05

99.0

699

.46

93.8

21.

001

NM

10.2

140.0

0-9

.99

139.

6614

0.11

93.5

01.

001

99.2

5-6

.05

99.0

799

.46

93.8

61.

001

NM

20.1

140.1

8-9

.88

139.

8414

0.37

93.5

21.

003

99.3

5-5

.87

99.1

999

.66

94.0

01.

003

NM

20.2

140.1

8-9

.93

139.

8514

0.40

93.5

81.

003

99.4

0-5

.91

99.2

499

.68

93.9

61.

003

NM

30.1

140.8

3-9

.56

140.

5214

0.77

93.4

41.

006

99.7

1-5

.61

99.5

699

.94

93.8

81.

005

NM

30.2

140.9

7-9

.59

140.

6614

0.89

93.4

01.

006

99.8

6-5

.63

99.7

110

0.03

93.9

61.

006

NM

40.1

141.2

1-9

.30

140.

9214

1.23

93.5

81.

009

99.4

5-5

.89

99.2

910

0.21

93.9

81.

008

NM

40.2

141.6

2-9

.34

141.

3214

1.53

93.5

01.

011

99.6

6-6

.11

99.4

810

0.41

94.0

21.

010

NM

50.1

141.8

2-9

.10

141.

5414

1.69

93.5

01.

012

100.

19-5

.68

100.

0410

0.54

93.8

21.

011

NM

50.2

142.7

9-9

.09

142.

5114

2.34

93.6

01.

017

100.

78-5

.71

100.

6310

1.00

94.0

81.

016

NM

60.1

141.8

5-9

.14

141.

5714

2.08

93.3

61.

015

100.

45-5

.11

100.

3310

0.88

93.9

61.

015

NM

60.2

143.2

9-9

.29

143.

0014

3.27

93.1

41.

023

101.

38-5

.36

101.

2410

1.72

94.1

81.

023

NM

70.1

142.5

2-8

.60

142.

2814

2.52

93.6

21.

018

100.

17-5

.71

100.

0210

1.07

93.9

21.

017

NM

70.2

144.7

2-9

.12

144.

4514

4.50

93.6

01.

032

101.

71-6

.37

101.

5210

2.47

94.0

61.

031

NM

80.1

141.8

4-8

.90

141.

5714

2.79

93.8

01.

020

100.

92-5

.13

100.

8010

1.35

93.8

41.

020

NM

80.2

144.5

2-9

.35

144.

2414

5.96

93.8

81.

043

103.

38-5

.04

103.

2710

3.67

93.8

21.

043

NM

90.1

142.4

5-8

.36

142.

2214

3.12

93.7

01.

022

100.

83-5

.38

100.

7010

1.49

93.9

61.

021

NM

90.2

147.4

6-8

.60

147.

2314

7.79

93.8

41.

056

104.

36-5

.85

104.

2110

4.81

94.2

41.

054

103

Tab

leE

44:

Infe

ren

cefo

rth

elo

gsc

ale

vari

anceσ

2b

ased

onLN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

45.2

8-1

.43

45.2

644

.66

94.1

81.

000

31.6

8-0

.28

31.6

931

.61

95.1

41.

000

MI.

C90

48.1

9-2

.18

48.1

444

.71

92.6

81.

001

33.4

6-0

.42

33.4

631

.67

93.5

01.

002

MI.

C80

49.0

9-0

.83

49.0

944

.85

92.3

01.

004

34.3

7-0

.05

34.3

831

.74

92.6

81.

004

MI.

D90

45.3

8-0

.76

45.3

844

.80

94.4

21.

003

31.7

40.

0731

.75

31.7

095

.20

1.00

3M

I.D

80

45.4

1-0

.66

45.4

144

.87

94.4

01.

005

31.8

90.

1431

.90

31.7

595

.14

1.00

4C

EN

S47

.11

-1.2

247

.10

46.5

094

.60

1.04

132

.75

0.11

32.7

532

.92

95.2

41.

041

NM

10.

145

.34

-1.4

245

.32

44.6

994

.22

1.00

131

.72

-0.2

631

.72

31.6

495

.14

1.00

1N

M10.

245

.35

-1.4

245

.33

44.6

994

.24

1.00

131

.72

-0.2

631

.72

31.6

495

.16

1.00

1N

M20.

145

.41

-1.4

245

.39

44.7

794

.18

1.00

331

.77

-0.2

431

.77

31.6

995

.20

1.00

3N

M20.

245

.43

-1.4

245

.42

44.7

894

.08

1.00

331

.77

-0.2

531

.77

31.7

095

.18

1.00

3N

M30.

145

.48

-1.4

345

.47

44.8

994

.36

1.00

531

.82

-0.2

531

.82

31.7

895

.12

1.00

5N

M30.

245

.55

-1.4

245

.54

44.9

294

.34

1.00

631

.84

-0.2

431

.85

31.8

095

.12

1.00

6N

M40.

145

.68

-1.3

245

.67

45.0

294

.08

1.00

831

.91

-0.2

331

.91

31.8

795

.22

1.00

8N

M40.

245

.75

-1.3

245

.74

45.1

294

.06

1.01

031

.99

-0.2

431

.99

31.9

495

.06

1.01

0N

M50.

145

.84

-1.3

245

.83

45.1

694

.30

1.01

131

.95

-0.1

431

.96

31.9

795

.00

1.01

1N

M50.

246

.07

-1.3

646

.06

45.3

794

.34

1.01

632

.19

-0.1

732

.19

32.1

294

.82

1.01

6N

M60.

145

.97

-1.4

145

.95

45.2

894

.12

1.01

431

.94

-0.1

331

.95

32.0

695

.20

1.01

4N

M60.

246

.36

-1.4

946

.34

45.6

794

.22

1.02

332

.17

-0.0

532

.18

32.3

495

.08

1.02

3N

M70.

145

.92

-1.5

645

.89

45.3

994

.22

1.01

632

.13

-0.0

232

.14

32.1

495

.10

1.01

7N

M70.

246

.59

-1.5

446

.57

46.0

594

.32

1.03

132

.66

-0.0

632

.66

32.6

195

.20

1.03

2N

M80.

146

.15

-1.3

546

.13

45.4

994

.42

1.01

932

.36

-0.1

332

.36

32.2

194

.86

1.01

9N

M80.

247

.20

-1.4

247

.19

46.5

494

.12

1.04

233

.25

-0.1

333

.25

32.9

594

.72

1.04

2N

M90.

146

.02

-1.3

746

.01

45.5

794

.50

1.02

032

.22

-0.0

632

.23

32.2

695

.12

1.02

0N

M90.

247

.59

-1.5

547

.57

47.0

994

.14

1.05

433

.38

-0.0

433

.38

33.3

594

.98

1.05

5

104

Tab

leE

45:

Infe

ren

cefo

rth

em

eaneµ

+σ2/2

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

199

.62

2.6

419

9.62

201.

9993

.52

1.00

014

1.26

-0.0

414

1.28

142.

5994

.68

1.00

0M

I.C

90

212

.68

-0.3

321

2.70

202.

8892

.60

1.00

414

8.84

-3.0

814

8.82

142.

6893

.68

1.00

1M

I.C

80

223

.29

12.1

022

2.98

207.

4192

.50

1.02

715

3.52

4.16

153.

4814

4.50

93.3

21.

013

MI.

D90

2×

1015

3×

1013

2×

1015

9×

1014

95.0

06×

1012

MI.

D80

230

.21

18.0

122

9.53

219.

2694

.36

1.08

614

3.77

6.05

143.

6614

5.09

95.0

61.

018

CE

NS

209

.04

7.6

420

8.92

209.

5493

.98

1.03

714

6.38

2.09

146.

3814

7.15

94.6

61.

032

NM

10.1

199

.80

2.7

419

9.80

202.

1293

.60

1.00

114

1.38

-0.1

114

1.40

142.

6594

.62

1.00

0N

M10

.2199

.77

2.7

219

9.78

202.

1293

.60

1.00

114

1.39

-0.1

114

1.41

142.

6594

.68

1.00

0N

M20

.1200

.13

2.8

820

0.13

202.

4193

.58

1.00

214

1.49

0.11

141.

5014

2.87

94.7

21.

002

NM

20.2

200

.14

2.8

220

0.14

202.

4493

.72

1.00

214

1.56

0.08

141.

5714

2.89

94.9

01.

002

NM

30.1

200

.74

3.3

320

0.74

202.

9093

.64

1.00

514

1.89

0.45

141.

9114

3.19

94.7

01.

004

NM

30.2

200

.95

3.3

220

0.95

203.

0593

.66

1.00

514

2.10

0.45

142.

1214

3.30

94.6

21.

005

NM

40.1

201

.39

3.6

820

1.38

203.

4393

.78

1.00

714

1.80

0.11

141.

8114

3.43

94.9

41.

006

NM

40.2

201

.81

3.6

820

1.79

203.

8293

.72

1.00

914

1.96

-0.1

714

1.97

143.

6694

.74

1.00

8N

M50

.1202

.30

4.0

320

2.28

203.

9693

.58

1.01

014

2.63

0.44

142.

6414

3.80

94.6

41.

008

NM

50.2

203

.36

4.1

720

3.34

204.

8493

.86

1.01

414

3.38

0.46

143.

3914

4.40

94.5

21.

013

NM

60.1

202

.98

4.0

420

2.96

204.

3893

.70

1.01

214

2.90

1.11

142.

9114

4.20

94.8

81.

011

NM

60.2

204

.99

4.1

020

4.97

206.

0293

.54

1.02

014

4.02

0.88

144.

0314

5.29

94.7

21.

019

NM

70.1

203

.55

4.7

220

3.52

204.

9393

.88

1.01

514

2.89

0.40

142.

9114

4.33

94.7

81.

012

NM

70.2

206

.80

4.4

220

6.78

207.

6993

.98

1.02

814

4.93

-0.3

214

4.95

146.

1994

.56

1.02

5N

M80

.1202

.86

4.2

820

2.84

205.

1493

.92

1.01

614

3.96

1.19

143.

9714

4.69

94.6

81.

015

NM

80.2

206

.98

4.1

420

6.96

209.

7793

.84

1.03

914

7.32

1.67

147.

3214

8.04

94.6

21.

038

NM

90.1

203

.19

4.9

820

3.14

205.

5893

.84

1.01

814

3.37

0.85

143.

3814

4.80

94.7

81.

015

NM

90.2

210

.88

5.5

221

0.83

213.

1494

.10

1.05

514

8.01

0.63

148.

0314

9.96

94.7

61.

052

105

Tab

leE

46:

Infe

ren

cefo

rth

em

eaneµ

+σ2/2

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

63.7

9-0

.65

63.7

963

.80

94.8

81.

000

45.1

40.

2845

.14

45.1

695

.22

1.00

0M

I.C

90

66.8

2-1

.43

66.8

163

.86

93.8

21.

001

47.0

80.

1547

.08

45.2

393

.86

1.00

1M

I.C

80

68.9

60.

6468

.96

64.1

493

.56

1.00

548

.35

0.73

48.3

545

.34

93.4

61.

004

MI.

D90

63.9

80.

1363

.98

64.0

295

.00

1.00

345

.22

0.68

45.2

245

.28

95.3

01.

003

MI.

D80

64.0

50.

4064

.06

64.1

895

.24

1.00

645

.34

0.85

45.3

445

.37

95.3

61.

005

CE

NS

65.5

6-0

.33

65.5

765

.62

95.0

21.

028

46.1

90.

7646

.19

46.4

595

.48

1.02

9N

M10.

163

.83

-0.6

363

.84

63.8

494

.78

1.00

145

.16

0.30

45.1

645

.19

95.2

21.

001

NM

10.

263

.84

-0.6

363

.85

63.8

494

.82

1.00

145

.16

0.30

45.1

745

.19

95.1

81.

001

NM

20.

163

.92

-0.6

363

.93

63.9

294

.96

1.00

245

.27

0.33

45.2

745

.25

95.1

21.

002

NM

20.

263

.96

-0.6

363

.96

63.9

394

.96

1.00

245

.28

0.32

45.2

845

.26

95.2

01.

002

NM

30.

163

.96

-0.6

463

.96

64.0

394

.94

1.00

445

.27

0.32

45.2

845

.33

95.1

21.

004

NM

30.

264

.08

-0.6

264

.08

64.0

895

.02

1.00

445

.36

0.32

45.3

645

.36

95.1

61.

004

NM

40.

164

.23

-0.5

064

.23

64.1

894

.82

1.00

645

.34

0.34

45.3

545

.42

95.2

81.

006

NM

40.

264

.41

-0.4

864

.41

64.3

094

.78

1.00

845

.46

0.32

45.4

745

.51

95.1

41.

008

NM

50.

164

.25

-0.5

064

.25

64.3

194

.80

1.00

845

.42

0.44

45.4

245

.52

95.2

81.

008

NM

50.

264

.58

-0.5

464

.58

64.5

795

.10

1.01

245

.73

0.41

45.7

345

.71

95.2

21.

012

NM

60.

164

.56

-0.5

964

.56

64.4

394

.78

1.01

045

.38

0.47

45.3

945

.61

95.2

61.

010

NM

60.

265

.13

-0.6

765

.14

64.9

394

.98

1.01

845

.75

0.57

45.7

545

.98

95.2

41.

018

NM

70.

164

.46

-0.7

664

.46

64.5

394

.92

1.01

145

.65

0.60

45.6

545

.70

95.4

41.

012

NM

70.

265

.34

-0.7

065

.34

65.4

294

.94

1.02

546

.35

0.55

46.3

546

.32

95.0

01.

026

NM

80.

164

.61

-0.5

264

.62

64.6

394

.88

1.01

345

.82

0.47

45.8

245

.75

95.1

01.

013

NM

80.

266

.02

-0.5

766

.02

66.1

095

.20

1.03

647

.12

0.51

47.1

246

.79

95.0

21.

036

NM

90.

164

.45

-0.5

564

.46

64.7

194

.86

1.01

445

.71

0.55

45.7

145

.81

95.2

81.

014

NM

90.

266

.79

-0.6

966

.80

67.0

295

.10

1.05

047

.30

0.63

47.3

047

.46

95.3

61.

051

106

Tab

leE

47:

Infe

ren

cefo

rth

eva

rian

cee2µ

+2σ2−e2µ

+σ2

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103

×10

3×

103

×10

3%

Len

.

CD

2169.1

4235

.84

2156

.49

2095

.21

89.6

81.

000

1432

.47

98.9

814

29.1

914

20.8

492

.00

1.00

0M

I.C

90

2540.5

4336

.20

2518

.44

2255

.95

87.7

81.

077

1578

.45

117.

0715

74.2

614

56.2

990

.86

1.02

5M

I.C

80

2898.6

5611

.67

2833

.66

2541

.08

89.0

21.

213

1693

.70

248.

0516

75.6

115

36.7

391

.24

1.08

2M

I.D

90

5×

1063

7×

1061

5×

1063

8×

1063

93.2

46×

1060

MI.

D80

5×

109

7×

107

5×

109

2×

109

92.2

08×

105

1555

.37

261.

4315

33.4

015

47.2

793

.50

1.08

9C

EN

S24

22.4

4335

.13

2399

.39

2261

.60

89.6

61.

079

1543

.33

139.

5015

37.1

715

03.8

692

.08

1.05

8N

M10.

121

73.3

5237

.68

2160

.53

2097

.81

89.6

41.

001

1435

.22

98.4

914

31.9

814

21.7

692

.14

1.00

1N

M10.

221

72.6

5237

.46

2159

.85

2097

.75

89.6

41.

001

1435

.38

98.5

514

32.1

314

21.8

292

.16

1.00

1N

M20.

121

79.6

1240

.46

2166

.52

2103

.56

89.8

81.

004

1437

.48

101.

4714

34.0

314

25.6

692

.16

1.00

3N

M20.

221

78.2

8239

.79

2165

.25

2103

.76

89.7

01.

004

1438

.16

101.

2014

34.7

414

25.9

692

.22

1.00

4N

M30.

121

96.7

1248

.98

2182

.78

2114

.49

89.6

41.

009

1446

.89

106.

7914

43.0

914

31.7

292

.22

1.00

8N

M30.

221

97.2

4249

.41

2183

.26

2116

.83

89.6

21.

010

1450

.00

107.

3114

46.1

714

33.3

892

.10

1.00

9N

M40.

122

21.4

2256

.10

2206

.83

2125

.67

89.7

01.

015

1443

.16

102.

1814

39.6

814

34.6

192

.36

1.01

0N

M40.

222

23.0

5257

.53

2208

.30

2131

.94

89.5

01.

018

1444

.62

99.4

714

41.3

414

37.3

892

.32

1.01

2N

M50.

122

34.0

4263

.15

2218

.71

2137

.06

89.5

01.

020

1462

.05

108.

5114

58.1

614

41.9

892

.28

1.01

5N

M50.

222

48.2

9268

.95

2232

.36

2152

.16

89.4

41.

027

1473

.97

110.

7014

69.9

514

50.8

192

.14

1.02

1N

M60.

122

47.4

0264

.79

2231

.97

2145

.14

89.6

01.

024

1465

.97

117.

4214

61.4

014

49.7

992

.42

1.02

0N

M60.

222

73.7

0272

.38

2257

.56

2171

.73

89.3

01.

037

1479

.14

117.

6014

74.6

114

64.9

192

.16

1.03

1N

M70.

122

64.3

2276

.41

2247

.61

2158

.25

89.5

81.

030

1464

.57

108.

4414

60.6

914

50.4

292

.22

1.02

1N

M70.

223

19.4

8284

.75

2302

.16

2202

.15

89.6

41.

051

1495

.50

105.

6514

91.9

114

75.5

392

.24

1.03

8N

M80.

122

49.0

1267

.85

2233

.23

2158

.77

89.7

01.

030

1486

.13

120.

6714

81.3

714

58.7

292

.06

1.02

7N

M80.

223

21.5

3280

.43

2304

.76

2228

.80

89.6

01.

064

1530

.57

133.

8415

24.8

615

05.9

592

.04

1.06

0N

M90.

122

63.3

8279

.01

2246

.34

2169

.82

89.3

81.

036

1475

.12

115.

5214

70.7

414

59.4

792

.26

1.02

7N

M90.

224

08.4

9313

.34

2388

.26

2286

.69

89.4

81.

091

1543

.02

124.

9015

38.1

115

27.5

091

.94

1.07

5

107

Tab

leE

48:

Infe

ren

cefo

rth

eva

rian

cee2µ

+2σ2−e2µ

+σ2

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le)

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

10

3×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

624

.10

15.2

562

3.97

618.

3894

.00

1.00

043

7.12

14.1

143

6.94

436.

7594

.88

1.00

0M

I.C

90

671

.08

15.8

567

0.96

621.

4892

.68

1.00

546

4.59

17.2

446

4.31

438.

5493

.20

1.00

4M

I.C

80

697

.96

46.8

669

6.46

629.

7292

.06

1.01

848

2.56

27.8

348

1.80

441.

3492

.50

1.01

1M

I.D

90

630

.13

32.9

262

9.33

625.

0694

.44

1.01

143

9.55

22.8

543

9.00

439.

5994

.80

1.00

7M

I.D

80

633

.04

41.6

663

1.73

629.

4394

.38

1.01

844

2.51

27.6

544

1.68

441.

5994

.88

1.01

1C

EN

S652

.93

21.5

365

2.64

646.

3194

.00

1.04

545

3.60

20.8

645

3.17

456.

3394

.96

1.04

5N

M10.

1625

.18

15.5

462

5.05

618.

8994

.00

1.00

143

7.54

14.3

043

7.35

437.

0994

.80

1.00

1N

M10.

2625

.34

15.6

562

5.21

618.

9294

.00

1.00

143

7.56

14.3

143

7.36

437.

1194

.82

1.00

1N

M20.

1626

.26

15.6

662

6.13

620.

1194

.06

1.00

343

8.75

14.7

943

8.54

437.

9994

.72

1.00

3N

M20.

2626

.64

15.7

362

6.51

620.

2993

.98

1.00

343

8.80

14.6

643

8.60

438.

1094

.74

1.00

3N

M30.

1627

.03

15.6

862

6.90

621.

8394

.04

1.00

643

9.25

14.7

043

9.05

439.

1994

.78

1.00

6N

M30.

2628

.44

15.9

662

8.30

622.

5094

.06

1.00

744

0.03

14.7

843

9.82

439.

6494

.76

1.00

7N

M40.

1630

.79

17.5

763

0.61

624.

1094

.08

1.00

944

0.39

15.0

144

0.18

440.

6194

.88

1.00

9N

M40.

2632

.66

17.9

363

2.47

625.

7893

.88

1.01

244

1.81

14.9

244

1.60

441.

7494

.68

1.01

1N

M50.

1632

.28

17.7

563

2.10

626.

1394

.24

1.01

344

1.45

16.3

144

1.19

442.

1694

.86

1.01

2N

M50.

2636

.28

17.6

363

6.10

629.

5694

.02

1.01

844

5.42

16.2

244

5.17

444.

5994

.60

1.01

8N

M60.

1636

.10

16.9

463

5.94

627.

8794

.14

1.01

544

1.08

16.5

444

0.81

443.

5094

.94

1.01

5N

M60.

2643

.58

16.6

664

3.43

634.

3393

.96

1.02

644

5.45

18.0

244

5.13

448.

2595

.00

1.02

6N

M70.

1634

.03

14.7

463

3.92

629.

1594

.26

1.01

744

4.78

18.3

144

4.44

444.

8594

.80

1.01

9N

M70.

2645

.71

16.3

964

5.56

640.

6494

.16

1.03

645

3.63

18.3

945

3.30

452.

7794

.70

1.03

7N

M80.

1638

.29

17.9

563

8.10

631.

0494

.02

1.02

044

7.69

16.9

844

7.41

445.

6694

.84

1.02

0N

M80.

2657

.02

19.0

865

6.81

649.

3993

.90

1.05

046

3.34

18.4

446

3.02

458.

6394

.48

1.05

0N

M90.

1635

.55

17.4

163

5.38

632.

0593

.94

1.02

244

5.44

17.8

044

5.13

446.

5294

.76

1.02

2N

M90.

2663

.99

18.2

766

3.80

659.

7894

.18

1.06

746

5.53

19.9

146

5.15

466.

2394

.64

1.06

8

108

Tab

leE

49:

Infe

ren

cefo

rth

e84

thp

erce

nti

leeµ

+σ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

327.7

9-2

.39

327.

8133

1.76

93.4

81.

000

232.

57-3

.43

232.

5723

4.68

94.6

81.

000

MI.

C90

347.9

0-9

.67

347.

8033

2.28

92.4

01.

002

244.

73-9

.48

244.

5723

4.62

93.6

01.

000

MI.

C80

363.2

58.

9236

3.17

338.

1992

.26

1.01

925

1.81

1.59

251.

8323

7.10

93.2

61.

010

MI.

D90

2×

1095

3×

1093

2×

1095

7×

1095

95.2

22×

1093

867.

0515

.49

867.

0030

0.80

95.0

01.

282

MI.

D80

335.5

316

.89

335.

1434

1.46

94.2

21.

029

235.

795.

1323

5.76

238.

0194

.88

1.01

4C

EN

S34

2.0

55.

1934

2.04

343.

4994

.04

1.03

524

0.56

-0.2

124

0.58

241.

9994

.62

1.03

1N

M10.

132

8.0

7-2

.24

328.

0933

1.96

93.4

81.

001

232.

75-3

.55

232.

7523

4.78

94.6

01.

000

NM

10.

232

8.0

3-2

.26

328.

0533

1.96

93.4

81.

001

232.

77-3

.55

232.

7723

4.79

94.6

21.

000

NM

20.

132

8.5

9-2

.03

328.

6133

2.43

93.6

41.

002

232.

92-3

.19

232.

9223

5.14

94.7

41.

002

NM

20.

232

8.6

0-2

.12

328.

6333

2.48

93.6

81.

002

233.

04-3

.25

233.

0423

5.18

94.8

41.

002

NM

30.

132

9.4

9-1

.35

329.

5233

3.19

93.6

41.

004

233.

54-2

.66

233.

5523

5.65

94.5

41.

004

NM

30.

232

9.8

6-1

.38

329.

8933

3.44

93.6

01.

005

233.

89-2

.67

233.

9023

5.82

94.5

81.

005

NM

40.

133

0.4

4-0

.81

330.

4733

4.01

93.8

61.

007

233.

40-3

.19

233.

4023

6.06

94.9

21.

006

NM

40.

233

1.1

8-0

.85

331.

2133

4.66

93.6

41.

009

233.

68-3

.68

233.

6723

6.45

94.7

81.

008

NM

50.

133

1.9

1-0

.29

331.

9533

4.85

93.6

61.

009

234.

70-2

.71

234.

7123

6.63

94.5

41.

008

NM

50.

233

3.6

6-0

.16

333.

6933

6.28

93.8

81.

014

235.

92-2

.72

235.

9323

7.62

94.4

41.

012

NM

60.

133

2.9

9-0

.29

333.

0333

5.53

93.5

81.

011

235.

13-1

.62

235.

1523

7.27

94.8

61.

011

NM

60.

233

6.3

2-0

.32

336.

3533

8.18

93.5

01.

019

237.

02-2

.05

237.

0323

9.07

94.6

81.

019

NM

70.

133

3.8

10.

7833

3.84

336.

3793

.74

1.01

423

5.13

-2.7

723

5.14

237.

5094

.64

1.01

2N

M70.

233

9.0

40.

0533

9.08

340.

8793

.90

1.02

723

8.47

-4.0

723

8.46

240.

5894

.50

1.02

5N

M80.

133

2.7

50.

1133

2.78

336.

7393

.78

1.01

523

6.79

-1.5

223

6.81

238.

0594

.64

1.01

4N

M80.

233

9.4

3-0

.39

339.

4634

4.29

93.9

01.

038

242.

35-0

.91

242.

3724

3.53

94.5

01.

038

NM

90.

133

3.1

61.

2033

3.19

337.

4093

.86

1.01

723

5.89

-2.0

723

5.90

238.

2494

.78

1.01

5N

M90.

234

5.5

01.

5834

5.53

349.

6394

.12

1.05

424

3.52

-2.7

024

3.53

246.

7094

.68

1.05

1

109

Tab

leE

50:

Infe

ren

cefo

rth

e84

thp

erce

nti

leeµ

+σ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

105

.15

-1.7

710

5.14

105.

1694

.88

1.00

074

.40

0.12

74.4

174

.45

95.2

01.

000

MI.

C90

110

.11

-3.2

611

0.07

105.

2493

.86

1.00

177

.61

-0.1

877

.61

74.5

493

.88

1.00

1M

I.C

80

113

.57

-0.0

111

3.58

105.

6593

.52

1.00

579

.67

0.68

79.6

874

.71

93.4

61.

004

MI.

D90

105

.41

-0.6

410

5.42

105.

4894

.92

1.00

374

.52

0.70

74.5

374

.62

95.3

01.

002

MI.

D80

105

.52

-0.2

910

5.53

105.

7295

.12

1.00

574

.72

0.94

74.7

274

.77

95.3

61.

004

CE

NS

108

.04

-1.3

010

8.04

108.

1495

.00

1.02

876

.13

0.89

76.1

376

.56

95.4

61.

028

NM

10.

1105

.22

-1.7

410

5.21

105.

2194

.78

1.00

174

.44

0.14

74.4

574

.48

95.1

61.

001

NM

10.

2105

.23

-1.7

310

5.23

105.

2294

.80

1.00

174

.45

0.15

74.4

574

.48

95.2

21.

001

NM

20.

1105

.37

-1.7

510

5.37

105.

3594

.98

1.00

274

.61

0.20

74.6

274

.58

95.1

21.

002

NM

20.

2105

.42

-1.7

410

5.42

105.

3795

.06

1.00

274

.63

0.18

74.6

474

.60

95.1

81.

002

NM

30.

1105

.43

-1.7

510

5.42

105.

5494

.96

1.00

474

.62

0.18

74.6

374

.72

95.1

61.

004

NM

30.

2105

.62

-1.7

310

5.62

105.

6294

.98

1.00

474

.76

0.19

74.7

774

.77

95.1

41.

004

NM

40.

1105

.86

-1.5

410

5.86

105.

7794

.78

1.00

674

.74

0.21

74.7

574

.87

95.3

21.

006

NM

40.

2106

.16

-1.5

010

6.16

105.

9894

.82

1.00

874

.94

0.18

74.9

575

.01

95.1

41.

008

NM

50.

1105

.90

-1.5

410

5.90

105.

9994

.80

1.00

874

.87

0.39

74.8

775

.03

95.3

01.

008

NM

50.

2106

.44

-1.6

210

6.44

106.

4395

.08

1.01

275

.37

0.33

75.3

875

.34

95.1

41.

012

NM

60.

1106

.40

-1.6

910

6.40

106.

1994

.80

1.01

074

.81

0.42

74.8

175

.18

95.2

61.

010

NM

60.

2107

.34

-1.8

310

7.33

107.

0194

.96

1.01

875

.41

0.59

75.4

275

.78

95.2

41.

018

NM

70.

1106

.25

-1.9

710

6.24

106.

3594

.98

1.01

175

.24

0.63

75.2

575

.32

95.4

41.

012

NM

70.

2107

.70

-1.9

010

7.70

107.

8194

.92

1.02

576

.39

0.55

76.4

076

.35

95.0

41.

026

NM

80.

1106

.48

-1.5

910

6.48

106.

5294

.84

1.01

375

.52

0.41

75.5

275

.41

95.0

81.

013

NM

80.

2108

.79

-1.7

010

8.79

108.

9395

.10

1.03

677

.67

0.47

77.6

877

.13

95.0

61.

036

NM

90.

1106

.23

-1.6

210

6.23

106.

6494

.84

1.01

475

.34

0.56

75.3

475

.51

95.2

81.

014

NM

90.

2110

.09

-1.9

011

0.08

110.

4595

.10

1.05

077

.97

0.66

77.9

778

.22

95.3

61.

051

110

Tab

leE

51:

Infe

ren

cefo

rth

e95

thp

erce

nti

leeµ

+1.6

45σ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103

×10

3×

103

×10

3%

Len

.

CD

787.

63-7

.93

787.

6779

3.99

93.1

41.

000

555.

13-9

.97

555.

1056

0.75

94.3

41.

000

MI.

C90

851.

74-2

0.91

851.

5779

8.26

91.6

81.

005

592.

52-2

3.34

592.

1256

1.40

92.8

01.

001

MI.

C80

896.

6233.4

289

6.08

818.

2291

.52

1.03

161

3.95

8.56

613.

9556

9.51

92.8

41.

016

MI.

D90

2205

24.6

031

32.4

722

0524

.41

2146

7.98

94.7

838

.284

MI.

D80

820.

2858.7

181

8.26

831.

9693

.92

1.04

856

6.91

18.9

456

6.65

572.

1094

.64

1.02

0C

EN

S833.

3814.5

583

3.34

832.

0393

.24

1.04

858

0.85

-0.4

258

0.91

584.

2994

.36

1.04

2N

M10.1

788.

48-7

.49

788.

5279

4.61

93.2

01.

001

555.

76-1

0.31

555.

7256

1.07

94.2

21.

001

NM

10.2

788.

39-7

.56

788.

4379

4.62

93.2

41.

001

555.

82-1

0.30

555.

7856

1.08

94.2

01.

001

NM

20.1

790.

00-6

.87

790.

0579

6.13

93.1

01.

003

556.

31-9

.29

556.

2956

2.19

94.4

41.

003

NM

20.2

790.

01-7

.11

790.

0579

6.28

93.0

81.

003

556.

64-9

.46

556.

6156

2.31

94.4

61.

003

NM

30.1

793.

35-4

.86

793.

4179

8.57

93.2

01.

006

558.

43-7

.77

558.

4456

3.82

94.3

41.

005

NM

30.2

794.

31-4

.91

794.

3779

9.31

93.0

41.

007

559.

46-7

.77

559.

4656

4.34

94.3

81.

006

NM

40.1

796.

58-3

.25

796.

6580

1.20

93.2

01.

009

557.

66-9

.28

557.

6356

5.12

94.2

81.

008

NM

40.2

798.

71-3

.27

798.

7880

3.15

93.2

41.

012

558.

53-1

0.54

558.

4956

6.29

94.2

41.

010

NM

50.1

800.

78-1

.73

800.

8680

3.90

93.2

41.

012

561.

94-7

.85

561.

9556

6.99

94.2

41.

011

NM

50.2

806.

21-1

.16

806.

2980

8.18

93.0

81.

018

565.

62-7

.77

565.

6256

9.90

94.2

41.

016

NM

60.1

803.

41-1

.67

803.

4980

6.10

93.0

41.

015

563.

32-4

.77

563.

3656

9.00

94.3

21.

015

NM

60.2

812.

86-1

.44

812.

9481

3.96

92.7

21.

025

568.

81-5

.80

568.

8357

4.30

94.3

41.

024

NM

70.1

806.

541.

4480

6.62

808.

8193

.06

1.01

956

2.89

-7.9

956

2.89

569.

7794

.30

1.01

6N

M70.2

821.

890.

0182

1.97

822.

0193

.22

1.03

557

2.78

-11.

1857

2.73

578.

7594

.04

1.03

2N

M80.1

803.

00-0

.54

803.

0880

9.98

93.2

61.

020

568.

05-4

.44

568.

0957

1.53

94.2

41.

019

NM

80.2

821.

97-1

.28

822.

0583

1.81

93.2

81.

048

584.

05-2

.50

584.

1058

7.34

94.0

41.

047

NM

90.1

805.

142.

6080

5.21

812.

1193

.16

1.02

356

5.73

-5.9

956

5.76

572.

1694

.32

1.02

0N

M90.2

841.

104.

7984

1.17

846.

7593

.32

1.06

658

8.35

-7.0

858

8.36

595.

9494

.52

1.06

3

111

Tab

leE

52:

Infe

ren

cefo

rth

e95

thp

erce

nti

leeµ

+1.6

45σ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=5.

18=

95th

per

centi

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

251.7

6-4

.39

251.

7525

1.07

94.6

21.

000

177.

500.

4517

7.52

177.

7594

.94

1.00

0M

I.C

9026

7.4

4-7

.82

267.

3525

1.39

93.1

21.

001

187.

37-0

.06

187.

3917

8.07

93.4

21.

002

MI.

C80

276.4

41.

2327

6.46

252.

6792

.76

1.00

619

3.40

2.51

193.

4117

8.60

92.8

21.

005

MI.

D90

252.6

4-0

.60

252.

6725

2.11

94.6

41.

004

177.

922.

3717

7.92

178.

3195

.04

1.00

3M

I.D

8025

2.9

90.

6425

3.01

252.

8294

.72

1.00

717

8.63

3.17

178.

6217

8.74

94.9

81.

006

CE

NS

261.1

6-2

.95

261.

1726

0.61

94.6

61.

038

183.

052.

6418

3.05

184.

5295

.26

1.03

8N

M10

.125

2.0

3-4

.31

252.

0125

1.24

94.6

41.

001

177.

640.

5117

7.66

177.

8795

.08

1.00

1N

M10

.225

2.0

7-4

.27

252.

0625

1.25

94.6

01.

001

177.

660.

5217

7.67

177.

8795

.08

1.00

1N

M20

.125

2.4

6-4

.31

252.

4525

1.67

94.7

41.

002

178.

110.

6717

8.13

178.

1895

.00

1.00

2N

M20

.225

2.6

2-4

.29

252.

6025

1.73

94.7

41.

003

178.

160.

6217

8.17

178.

2294

.98

1.00

3N

M30

.125

2.7

1-4

.32

252.

7025

2.28

94.6

21.

005

178.

220.

6317

8.24

178.

6195

.02

1.00

5N

M30

.225

3.2

5-4

.26

253.

2425

2.51

94.6

21.

006

178.

560.

6417

8.58

178.

7795

.02

1.00

6N

M40

.125

4.0

0-3

.72

254.

0025

3.02

94.5

81.

008

178.

620.

7117

8.64

179.

1094

.90

1.00

8N

M40

.225

4.7

5-3

.63

254.

7525

3.63

94.6

21.

010

179.

180.

6417

9.20

179.

5394

.78

1.01

0N

M50

.125

4.3

5-3

.71

254.

3525

3.72

94.5

41.

011

178.

981.

1917

8.99

179.

6395

.12

1.01

1N

M50

.225

5.9

1-3

.90

255.

9025

5.00

94.6

61.

016

180.

471.

0618

0.48

180.

5495

.22

1.01

6N

M60

.125

5.6

8-4

.12

255.

6825

4.35

94.6

41.

013

178.

831.

2917

8.84

180.

1095

.10

1.01

3N

M60

.225

8.3

7-4

.48

258.

3625

6.77

94.7

81.

023

180.

511.

7618

0.52

181.

8595

.22

1.02

3N

M70

.125

5.2

2-4

.91

255.

2025

4.86

94.8

41.

015

180.

111.

8818

0.11

180.

5395

.26

1.01

6N

M70

.225

9.4

8-4

.66

259.

4625

9.13

94.7

01.

032

183.

461.

6918

3.47

183.

5394

.84

1.03

3N

M80

.125

6.1

8-3

.82

256.

1825

5.42

94.7

81.

017

181.

081.

3018

1.09

180.

8494

.92

1.01

7N

M80

.226

2.9

5-4

.03

262.

9526

2.34

94.8

01.

045

187.

151.

4818

7.16

185.

7694

.82

1.04

5N

M90

.125

5.3

9-3

.94

255.

3925

5.80

94.8

81.

019

180.

441.

6718

0.45

181.

1395

.16

1.01

9N

M90

.226

6.2

4-4

.58

266.

2326

6.47

94.9

61.

061

188.

032.

0018

8.04

188.

7595

.06

1.06

2

112

Tab

leE

53:

Infe

ren

cefo

rth

elo

gsc

ale

mea

nµ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

99.6

8-1

.73

99.6

899

.19

94.7

01.

000

71.8

70.

4171

.88

70.3

794

.36

1.00

0M

I.C

90

101

.42

-2.3

010

1.40

99.2

994

.30

1.00

173

.00

0.19

73.0

070

.42

94.1

21.

001

MI.

C80

105

.18

3.28

105.

1410

0.00

93.7

81.

008

74.8

72.

6874

.83

70.7

293

.60

1.00

5M

I.D

90

99.7

3-1

.73

99.7

399

.79

94.7

41.

006

71.9

90.

4072

.00

70.5

994

.32

1.00

3M

I.D

80

100

.04

-1.5

610

0.04

100.

0394

.76

1.00

872

.03

0.40

72.0

370

.74

94.5

21.

005

CE

NS

100

.80

-0.8

210

0.81

100.

4094

.80

1.01

272

.70

0.69

72.7

071

.12

94.6

61.

011

NM

10.

199.6

7-1

.69

99.6

699

.21

94.7

41.

000

71.8

70.

4271

.87

70.3

894

.48

1.00

0N

M10.

299.6

6-1

.69

99.6

699

.21

94.7

41.

000

71.8

60.

4271

.87

70.3

894

.46

1.00

0N

M20.

199.7

6-1

.69

99.7

599

.25

94.6

21.

001

71.9

10.

4171

.92

70.4

194

.42

1.00

1N

M20.

299.7

7-1

.67

99.7

799

.26

94.6

41.

001

71.9

10.

3971

.92

70.4

194

.42

1.00

1N

M30.

199.7

8-1

.79

99.7

899

.29

94.7

61.

001

71.9

50.

3771

.96

70.4

494

.42

1.00

1N

M30.

299.7

9-1

.81

99.7

899

.32

94.7

21.

001

71.9

30.

3571

.94

70.4

694

.38

1.00

1N

M40.

199.7

9-1

.78

99.7

899

.36

94.6

01.

002

72.0

00.

5372

.00

70.5

194

.42

1.00

2N

M40.

299.7

7-1

.83

99.7

799

.43

94.7

01.

002

72.0

50.

5172

.05

70.5

694

.40

1.00

3N

M50.

1100

.01

-1.7

310

0.01

99.4

494

.68

1.00

372

.11

0.52

72.1

270

.56

94.3

21.

003

NM

50.

2100

.12

-1.7

610

0.12

99.6

094

.70

1.00

472

.27

0.55

72.2

770

.68

94.3

81.

004

NM

60.

199.9

9-1

.43

99.9

999

.56

94.7

01.

004

72.0

50.

4772

.06

70.6

094

.32

1.00

3N

M60.

2100

.31

-1.4

310

0.31

99.9

094

.82

1.00

772

.26

0.41

72.2

670

.84

94.5

21.

007

NM

70.

199.9

7-1

.53

99.9

799

.60

94.7

01.

004

72.1

60.

4172

.16

70.6

394

.50

1.00

4N

M70.

2100

.45

-1.6

710

0.44

100.

2394

.74

1.01

072

.57

0.50

72.5

871

.10

94.4

61.

010

NM

80.

1100

.01

-1.4

710

0.01

99.6

794

.56

1.00

572

.27

0.47

72.2

770

.67

94.3

81.

004

NM

80.

2101

.11

-1.4

610

1.11

100.

8494

.82

1.01

772

.94

0.60

72.9

571

.53

94.4

41.

016

NM

90.

1100

.07

-1.5

410

0.07

99.7

094

.56

1.00

572

.24

0.45

72.2

570

.70

94.5

01.

005

NM

90.

2101

.74

-1.6

110

1.74

101.

8594

.82

1.02

773

.81

0.77

73.8

172

.28

94.8

41.

027

113

Tab

leE

54:

Infe

ren

cefo

rth

elo

gsc

ale

mea

nµ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

31.2

3-0

.02

31.2

331

.61

94.8

81.

000

22.5

40.

2022

.54

22.3

594

.78

1.00

0M

I.C

90

31.7

0-0

.05

31.7

131

.63

94.6

41.

001

22.9

20.

2322

.92

22.3

694

.54

1.00

1M

I.C

80

32.2

00.

4632

.20

31.6

994

.20

1.00

323

.34

0.51

23.3

322

.40

94.2

61.

002

MI.

D90

31.2

7-0

.07

31.2

731

.64

95.0

01.

001

22.5

70.

2022

.57

22.3

794

.78

1.00

1M

I.D

80

31.3

5-0

.01

31.3

531

.70

94.8

21.

003

22.5

60.

2022

.56

22.4

194

.94

1.00

3C

EN

S31

.58

0.16

31.5

831

.93

94.8

01.

010

22.7

70.

3222

.77

22.5

894

.86

1.01

0N

M10.

131

.23

-0.0

331

.23

31.6

194

.92

1.00

022

.54

0.19

22.5

522

.35

94.8

41.

000

NM

10.

231

.23

-0.0

331

.24

31.6

194

.94

1.00

022

.55

0.19

22.5

522

.35

94.7

81.

000

NM

20.

131

.24

-0.0

331

.24

31.6

295

.02

1.00

122

.56

0.21

22.5

622

.36

94.7

21.

001

NM

20.

231

.24

-0.0

431

.24

31.6

294

.94

1.00

122

.56

0.20

22.5

622

.36

94.7

61.

001

NM

30.

131

.27

0.00

31.2

731

.64

95.0

21.

001

22.5

70.

1822

.58

22.3

794

.76

1.00

1N

M30.

231

.26

-0.0

031

.27

31.6

595

.02

1.00

122

.59

0.19

22.5

922

.38

94.8

41.

001

NM

40.

131

.29

-0.0

131

.30

31.6

695

.04

1.00

222

.60

0.21

22.6

022

.39

94.7

41.

002

NM

40.

231

.31

-0.0

331

.31

31.6

995

.10

1.00

322

.63

0.21

22.6

322

.40

94.8

21.

003

NM

50.

131

.30

0.00

31.3

031

.69

95.0

41.

003

22.5

80.

1822

.58

22.4

095

.06

1.00

2N

M50.

231

.34

-0.0

431

.34

31.7

495

.10

1.00

422

.62

0.19

22.6

222

.44

94.8

41.

004

NM

60.

131

.35

0.02

31.3

631

.71

94.9

01.

003

22.6

00.

2222

.60

22.4

294

.94

1.00

3N

M60.

231

.43

-0.0

431

.44

31.8

295

.10

1.00

722

.70

0.19

22.7

122

.50

94.7

41.

007

NM

70.

131

.33

0.09

31.3

331

.73

94.9

41.

004

22.6

10.

2422

.61

22.4

394

.88

1.00

4N

M70.

231

.50

0.12

31.5

031

.94

94.9

81.

011

22.7

60.

2522

.76

22.5

895

.02

1.01

0N

M80.

131

.36

-0.0

031

.36

31.7

494

.92

1.00

422

.69

0.25

22.6

922

.44

94.8

01.

004

NM

80.

231

.77

-0.1

231

.77

32.1

295

.18

1.01

622

.92

0.25

22.9

222

.71

95.0

81.

016

NM

90.

131

.41

0.07

31.4

231

.76

94.9

21.

005

22.6

20.

2422

.62

22.4

594

.92

1.00

5N

M90.

232

.17

-0.0

032

.17

32.4

695

.04

1.02

723

.16

0.21

23.1

622

.95

94.9

21.

027

114

Tab

leE

55:

Infe

ren

cefo

rth

elo

gsc

ale

vari

anceσ

2b

ased

onLN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

140.

77-1

1.19

140.

3413

9.84

93.1

21.

000

98.4

8-7

.15

98.2

399

.28

94.4

81.

000

MI.

C90

154.

77-9

.50

154.

4914

1.11

91.0

41.

009

108.

43-6

.43

108.

2599

.89

92.2

41.

006

MI.

C80

156.

231.

7315

6.24

143.

6591

.50

1.02

710

9.33

-0.8

810

9.34

100.

9592

.12

1.01

7M

I.D

90

144.

43-1

.11

144.

4414

3.03

93.7

81.

023

99.1

3-2

.85

99.1

010

0.31

94.6

61.

010

MI.

D80

143.

42-0

.15

143.

4414

3.06

93.6

41.

023

99.4

2-1

.93

99.4

110

0.64

94.8

41.

014

CE

NS

151.

96-7

.03

151.

8115

2.37

93.4

61.

090

107.

77-5

.99

107.

6110

7.71

94.2

41.

085

NM

10.1

140.

95-1

0.97

140.

5314

0.01

93.0

81.

001

98.5

7-7

.12

98.3

299

.39

94.4

61.

001

NM

10.

2140.

95-1

0.98

140.

5314

0.02

93.0

41.

001

98.5

8-7

.13

98.3

399

.39

94.4

81.

001

NM

20.

1141.

55-1

1.03

141.

1314

0.40

93.0

21.

004

98.8

2-7

.19

98.5

799

.66

94.4

61.

004

NM

20.

2141.

59-1

0.98

141.

1814

0.44

93.0

01.

004

98.8

4-7

.22

98.5

899

.69

94.5

21.

004

NM

30.

1142.

08-1

1.28

141.

6514

0.95

93.1

61.

008

99.0

6-7

.33

98.8

010

0.07

94.5

81.

008

NM

30.

2142.

32-1

1.33

141.

8814

1.07

93.1

41.

009

99.2

1-7

.37

98.9

410

0.15

94.6

01.

009

NM

40.

1142.

25-1

1.15

141.

8314

1.67

93.0

01.

013

99.9

6-6

.70

99.7

410

0.64

94.2

81.

014

NM

40.

2142.

61-1

1.28

142.

1714

1.99

93.0

41.

015

100.

25-6

.75

100.

0310

0.87

94.2

61.

016

NM

50.1

142.

19-1

1.08

141.

7714

2.44

93.3

61.

019

100.

36-6

.79

100.

1410

1.17

94.5

41.

019

NM

50.2

143.

14-1

1.20

142.

7114

3.16

93.3

21.

024

100.

79-6

.72

100.

5810

1.70

94.2

61.

024

NM

60.1

144.

51-9

.78

144.

1914

3.39

93.2

61.

025

101.

22-6

.93

100.

9910

1.69

94.2

01.

024

NM

60.2

145.

89-9

.84

145.

5714

4.77

93.0

21.

035

102.

24-7

.09

102.

0010

2.66

93.8

81.

034

NM

70.1

144.

39-1

0.19

144.

0414

4.03

93.3

21.

030

101.

27-7

.18

101.

0310

2.16

94.3

41.

029

NM

70.2

147.

47-1

0.55

147.

1114

6.38

93.2

41.

047

102.

76-7

.02

102.

5310

3.90

94.5

41.

046

NM

80.1

144.

95-9

.84

144.

6314

4.70

93.2

61.

035

101.

50-7

.00

101.

2610

2.61

94.4

01.

033

NM

80.2

149.

14-9

.91

148.

8214

8.51

93.0

81.

062

104.

18-6

.75

103.

9710

5.38

94.5

41.

061

NM

90.1

144.

52-1

0.28

144.

1614

5.15

93.5

61.

038

102.

68-7

.03

102.

4510

2.97

94.2

01.

037

NM

90.2

149.

56-1

0.76

149.

1815

0.55

93.4

21.

077

106.

94-6

.63

106.

7410

6.95

94.0

21.

077

115

Tab

leE

56:

Infe

ren

cefo

rth

elo

gsc

ale

vari

anceσ

2b

ased

onLN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

44.7

8-0

.60

44.7

844

.69

94.6

61.

000

31.4

5-1

.04

31.4

431

.59

94.8

21.

000

MI.

C90

48.8

8-0

.24

48.8

944

.88

92.6

01.

004

34.2

8-0

.70

34.2

731

.71

92.7

41.

004

MI.

C80

49.2

70.

9549

.26

45.1

092

.64

1.00

934

.49

-0.1

034

.49

31.8

593

.08

1.00

8M

I.D

90

44.8

90.

0844

.89

44.9

194

.46

1.00

531

.60

-0.6

431

.59

31.7

294

.80

1.00

4M

I.D

80

45.1

50.

4045

.15

45.0

194

.58

1.00

731

.60

-0.5

531

.60

31.7

995

.00

1.00

6C

EN

S48

.41

0.18

48.4

248

.44

94.7

81.

084

34.0

9-0

.53

34.0

934

.22

94.7

41.

083

NM

10.

144

.82

-0.6

144

.82

44.7

494

.58

1.00

131

.47

-1.0

831

.46

31.6

294

.70

1.00

1N

M10.

244

.82

-0.6

244

.82

44.7

494

.56

1.00

131

.47

-1.0

831

.46

31.6

294

.70

1.00

1N

M20.

144

.97

-0.6

044

.97

44.8

794

.52

1.00

431

.61

-1.0

131

.60

31.7

194

.74

1.00

4N

M20.

245

.00

-0.6

345

.00

44.8

894

.56

1.00

431

.62

-1.0

231

.61

31.7

294

.72

1.00

4N

M30.

145

.07

-0.5

045

.07

45.0

694

.88

1.00

831

.78

-1.1

231

.76

31.8

494

.60

1.00

8N

M30.

245

.14

-0.5

245

.14

45.1

094

.82

1.00

931

.80

-1.1

031

.78

31.8

794

.64

1.00

9N

M40.

145

.37

-0.5

345

.37

45.2

994

.70

1.01

331

.90

-1.0

131

.89

32.0

194

.54

1.01

3N

M40.

245

.51

-0.5

745

.51

45.3

994

.70

1.01

631

.96

-1.0

231

.95

32.0

894

.64

1.01

6N

M50.

145

.49

-0.4

745

.49

45.5

494

.70

1.01

932

.02

-1.1

032

.01

32.1

894

.66

1.01

9N

M50.

245

.82

-0.5

745

.82

45.7

794

.66

1.02

432

.17

-1.0

832

.15

32.3

494

.80

1.02

4N

M60.

145

.78

-0.4

445

.79

45.7

894

.84

1.02

432

.30

-0.9

232

.29

32.3

594

.64

1.02

4N

M60.

246

.33

-0.5

846

.33

46.2

294

.66

1.03

432

.50

-1.0

032

.49

32.6

794

.72

1.03

4N

M70.

145

.94

-0.1

345

.94

46.0

294

.84

1.03

032

.39

-0.9

032

.38

32.5

194

.74

1.02

9N

M70.

246

.88

-0.0

646

.89

46.8

094

.82

1.04

733

.00

-0.8

732

.99

33.0

694

.76

1.04

7N

M80.

146

.13

-0.4

946

.13

46.1

994

.94

1.03

432

.60

-0.8

332

.59

32.6

594

.52

1.03

4N

M80.

247

.46

-0.7

847

.46

47.4

294

.98

1.06

133

.41

-0.8

233

.40

33.5

394

.72

1.06

1N

M90.

146

.16

-0.2

146

.17

46.3

794

.72

1.03

732

.69

-0.8

732

.68

32.7

694

.46

1.03

7N

M90.

248

.10

-0.4

248

.11

48.1

495

.02

1.07

733

.86

-0.9

533

.85

34.0

294

.70

1.07

7

116

Tab

leE

57:

Infe

ren

cefo

rth

em

eaneµ

+σ2/2

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

202.0

40.

2020

2.06

201.

5693

.24

1.00

014

3.17

0.96

143.

1814

2.56

94.3

61.

000

MI.

C90

222.9

36.

9122

2.85

206.

2692

.10

1.02

315

5.82

4.09

155.

7814

4.28

92.8

21.

012

MI.

C80

240.1

832

.76

237.

9621

5.24

92.3

01.

068

164.

3116

.03

163.

5414

7.52

92.6

41.

035

MI.

D90

6×

1021

9×

1019

6×

1021

5×

1021

93.9

02×

1019

145.

446.

9514

5.29

145.

0294

.60

1.01

7M

I.D

80

212.0

219

.65

211.

1321

2.08

94.1

01.

052

146.

569.

7514

6.25

146.

4295

.02

1.02

7C

EN

S21

7.1

06.

8021

7.02

216.

6693

.58

1.07

515

4.19

3.29

154.

1715

2.04

94.5

41.

067

NM

10.

120

2.1

90.

4720

2.21

201.

7993

.30

1.00

114

3.20

1.00

143.

2114

2.68

94.3

41.

001

NM

10.

220

2.1

80.

4720

2.20

201.

8093

.22

1.00

114

3.19

0.99

143.

2014

2.69

94.4

21.

001

NM

20.

120

3.0

50.

5220

3.07

202.

2593

.18

1.00

314

3.62

0.96

143.

6314

2.99

94.4

81.

003

NM

20.

220

3.1

30.

6120

3.15

202.

3393

.14

1.00

414

3.62

0.91

143.

6314

3.02

94.3

61.

003

NM

30.

120

3.4

90.

2220

3.51

202.

8593

.22

1.00

614

3.88

0.79

143.

9014

3.43

94.6

21.

006

NM

30.

220

3.6

90.

1720

3.71

203.

0493

.10

1.00

714

3.94

0.74

143.

9514

3.56

94.5

01.

007

NM

40.

120

3.6

10.

3520

3.63

203.

6793

.00

1.01

014

4.84

1.67

144.

8414

4.13

94.4

41.

011

NM

40.

220

3.8

20.

1820

3.84

204.

1793

.04

1.01

314

5.25

1.63

145.

2614

4.50

94.2

61.

014

NM

50.

120

4.5

50.

6020

4.57

204.

5893

.54

1.01

514

5.65

1.63

145.

6614

4.72

94.2

61.

015

NM

50.

220

5.6

80.

5820

5.70

205.

7493

.24

1.02

114

6.50

1.82

146.

5014

5.57

94.4

81.

021

NM

60.

120

6.6

82.

3720

6.69

205.

8893

.18

1.02

114

5.94

1.47

145.

9514

5.28

94.4

81.

019

NM

60.

220

9.0

22.

6020

9.03

208.

1893

.16

1.03

314

7.46

1.38

147.

4714

6.86

94.3

61.

030

NM

70.

120

6.4

81.

8620

6.49

206.

5393

.30

1.02

514

6.53

1.22

146.

5414

5.78

94.2

81.

023

NM

70.

221

0.5

41.

8421

0.55

210.

5793

.28

1.04

514

9.06

1.71

149.

0614

8.71

94.5

01.

043

NM

80.

120

7.0

92.

3020

7.10

207.

3493

.36

1.02

914

7.16

1.51

147.

1714

6.30

94.3

81.

026

NM

80.

221

4.3

83.

1421

4.38

214.

3393

.14

1.06

315

1.57

2.31

151.

5615

1.24

94.3

61.

061

NM

90.

120

6.9

21.

8120

6.93

207.

7993

.14

1.03

114

8.05

1.53

148.

0614

6.70

94.4

61.

029

NM

90.

221

6.0

12.

4321

6.02

218.

7493

.56

1.08

515

6.18

3.10

156.

1715

4.65

94.3

21.

085

117

Tab

leE

58:

Infe

ren

cefo

rth

em

eaneµ

+σ2/2

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

62.8

40.

6662

.84

63.8

995

.36

1.00

044

.83

0.08

44.8

445

.13

95.1

81.

000

MI.

C90

67.7

61.

4667

.75

64.2

093

.24

1.00

548

.34

0.68

48.3

445

.32

93.3

61.

004

MI.

C80

69.6

93.

8669

.59

64.6

993

.30

1.01

349

.67

1.91

49.6

445

.58

92.7

21.

010

MI.

D90

63.1

61.

5863

.14

64.2

595

.36

1.00

645

.09

0.63

45.0

945

.34

95.2

61.

005

MI.

D80

63.6

72.

3663

.63

64.5

695

.34

1.01

145

.10

0.90

45.1

045

.52

95.1

61.

009

CE

NS

66.9

71.

7666

.96

67.9

195

.28

1.06

347

.65

0.78

47.6

547

.94

94.9

41.

062

NM

10.

162

.88

0.64

62.8

863

.94

95.2

21.

001

44.8

70.

0444

.87

45.1

795

.18

1.00

1N

M10.

262

.89

0.64

62.8

963

.94

95.2

81.

001

44.8

70.

0444

.88

45.1

795

.16

1.00

1N

M20.

163

.03

0.66

63.0

364

.08

95.2

61.

003

45.0

20.

1245

.03

45.2

795

.14

1.00

3N

M20.

263

.04

0.62

63.0

464

.10

95.3

21.

003

45.0

30.

1145

.03

45.2

895

.20

1.00

3N

M30.

163

.22

0.80

63.2

264

.30

95.4

21.

006

45.2

0-0

.00

45.2

145

.41

95.1

81.

006

NM

30.

263

.27

0.78

63.2

764

.36

95.4

01.

007

45.2

60.

0245

.27

45.4

695

.02

1.00

7N

M40.

163

.54

0.77

63.5

464

.55

95.2

41.

010

45.4

00.

1445

.40

45.5

995

.06

1.01

0N

M40.

263

.70

0.72

63.7

064

.71

95.3

41.

013

45.5

50.

1345

.55

45.7

195

.20

1.01

3N

M50.

163

.65

0.84

63.6

564

.81

95.3

01.

014

45.4

00.

0245

.41

45.7

794

.98

1.01

4N

M50.

264

.05

0.71

64.0

565

.17

95.2

61.

020

45.6

40.

0645

.64

46.0

394

.98

1.02

0N

M60.

164

.09

0.90

64.0

965

.08

95.1

61.

019

45.6

90.

2445

.70

45.9

795

.18

1.01

8N

M60.

264

.81

0.73

64.8

165

.78

95.2

61.

030

46.1

70.

1446

.17

46.4

795

.44

1.03

0N

M70.

164

.16

1.28

64.1

565

.34

95.5

41.

023

45.8

00.

2845

.80

46.1

494

.98

1.02

2N

M70.

265

.50

1.45

65.4

966

.63

95.4

01.

043

46.7

90.

3646

.80

47.0

494

.88

1.04

2N

M80.

164

.37

0.85

64.3

765

.52

95.1

81.

025

46.2

40.

3846

.24

46.2

994

.88

1.02

6N

M80.

266

.68

0.51

66.6

967

.65

95.5

21.

059

47.6

10.

4247

.62

47.8

195

.22

1.05

9N

M90.

164

.61

1.20

64.6

065

.71

95.3

81.

029

46.0

20.

3346

.02

46.4

095

.22

1.02

8N

M90.

268

.31

1.06

68.3

169

.15

95.1

41.

082

48.4

80.

2748

.48

48.8

495

.20

1.08

2

118

Tab

leE

59:

Infe

ren

cefo

rth

eva

rian

cee2µ

+2σ2−e2µ

+σ2

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103

×10

3×

103

×10

3%

Len

.

CD

214

8.4

921

6.4

621

37.7

720

85.7

988

.92

1.00

014

27.9

795

.62

1424

.91

1418

.54

91.4

61.

000

MI.

C90

277

2.2

254

5.5

427

18.2

824

86.2

487

.92

1.19

216

97.1

523

4.11

1681

.10

1528

.93

90.9

01.

078

MI.

C80

329

9.2

1106

2.8

131

23.6

530

58.7

090

.64

1.46

619

03.7

645

3.04

1849

.25

1681

.33

92.5

01.

185

MI.

D90

6×

1089

9×

108

76×

1089

2×

1090

91.7

89×

1986

1533

.35

253.

9615

12.3

315

41.8

392

.92

1.08

7M

I.D

80273

8.9

279

1.5

926

22.3

027

87.0

792

.76

1.33

615

88.1

233

1.00

1553

.40

1607

.07

93.7

61.

133

CE

NS

249

3.0

335

7.9

424

67.4

523

88.7

888

.72

1.14

516

43.8

915

5.73

1636

.67

1581

.10

91.4

61.

115

NM

10.1

215

2.9

722

0.4

721

41.8

720

90.3

588

.70

1.00

214

28.6

996

.28

1425

.58

1420

.43

91.4

61.

001

NM

10.2

215

3.1

822

0.4

621

42.0

820

90.4

288

.72

1.00

214

28.5

696

.19

1425

.46

1420

.45

91.4

61.

001

NM

20.1

217

2.0

622

4.3

621

60.6

520

99.4

988

.88

1.00

714

36.5

596

.83

1433

.43

1425

.23

91.4

21.

005

NM

20.2

217

2.9

822

5.5

021

61.4

721

00.7

388

.86

1.00

714

36.8

796

.32

1433

.78

1425

.51

91.4

21.

005

NM

30.1

217

2.6

822

2.7

421

61.4

421

08.7

788

.78

1.01

114

37.6

195

.46

1434

.58

1431

.44

91.5

01.

009

NM

30.2

217

6.2

322

3.1

221

64.9

821

11.5

088

.64

1.01

214

39.6

495

.31

1436

.63

1433

.13

91.6

41.

010

NM

40.1

217

7.6

722

5.2

121

66.2

121

22.2

088

.86

1.01

714

58.2

010

8.58

1454

.30

1444

.58

90.9

81.

018

NM

40.2

217

9.8

522

4.5

721

68.4

721

28.3

888

.82

1.02

014

64.3

710

9.19

1460

.44

1449

.35

91.2

61.

022

NM

50.1

218

7.1

822

9.2

221

75.3

621

37.4

588

.88

1.02

514

71.2

110

9.80

1467

.25

1453

.91

91.4

01.

025

NM

50.2

220

3.9

823

3.2

921

91.8

221

53.9

488

.78

1.03

314

82.9

511

3.55

1478

.75

1465

.08

91.1

41.

033

NM

60.1

225

8.1

426

1.9

522

43.1

221

69.8

688

.78

1.04

014

75.8

510

9.76

1471

.91

1462

.72

91.4

01.

031

NM

60.2

229

6.6

727

2.1

222

80.7

222

02.9

088

.78

1.05

614

98.8

611

2.44

1494

.79

1482

.65

91.4

61.

045

NM

70.1

224

5.8

125

4.7

122

31.5

421

78.2

688

.94

1.04

414

88.1

810

7.81

1484

.42

1470

.23

91.5

41.

036

NM

70.2

231

4.2

127

0.3

322

98.5

922

36.0

788

.64

1.07

215

23.3

111

8.71

1518

.83

1507

.56

91.9

61.

063

NM

80.1

226

3.0

026

3.2

222

47.8

721

94.3

888

.88

1.05

214

95.2

411

2.27

1491

.17

1478

.98

91.5

41.

043

NM

80.2

238

4.5

629

6.8

823

66.2

422

93.9

988

.36

1.10

015

58.5

113

0.98

1553

.15

1541

.00

91.6

21.

086

NM

90.1

224

1.9

525

4.3

822

27.6

921

98.4

788

.96

1.05

415

20.0

511

6.21

1515

.75

1486

.90

91.5

61.

048

NM

90.2

237

2.4

028

8.4

923

55.0

323

39.6

788

.80

1.12

216

27.4

414

9.48

1620

.72

1583

.48

91.0

81.

116

119

Tab

leE

60:

Infe

ren

cefo

rth

eva

rian

cee2µ

+2σ2−e2µ

+σ2

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

10

3×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

617

.36

28.0

461

6.78

620.

3394

.74

1.00

043

1.88

6.94

431.

8743

5.86

94.8

61.

000

MI.

C90

689

.99

54.9

668

7.87

630.

9392

.62

1.01

747

9.65

22.5

147

9.17

440.

6392

.68

1.01

1M

I.C

80

715

.76

94.5

270

9.56

644.

4392

.82

1.03

949

2.50

42.1

149

0.75

446.

3093

.04

1.02

4M

I.D

90

626

.87

53.0

162

4.69

631.

1495

.06

1.01

743

6.98

20.2

043

6.56

440.

6695

.10

1.01

1M

I.D

80

636

.15

70.2

763

2.32

638.

4495

.08

1.02

943

8.18

27.3

143

7.37

443.

7395

.28

1.01

8C

EN

S678

.59

46.1

967

7.08

679.

8094

.74

1.09

647

2.95

17.7

547

2.66

476.

3294

.58

1.09

3N

M10.

1617

.98

27.9

261

7.41

621.

0394

.74

1.00

143

2.18

6.51

432.

1843

6.32

94.7

81.

001

NM

10.

2618

.10

27.8

461

7.53

621.

0494

.70

1.00

143

2.19

6.50

432.

1843

6.33

94.7

21.

001

NM

20.

1620

.05

28.3

161

9.47

623.

0794

.50

1.00

443

4.48

7.62

434.

4643

7.82

94.5

01.

005

NM

20.

2620

.33

27.9

061

9.77

623.

2094

.54

1.00

543

4.64

7.54

434.

6243

7.96

94.4

41.

005

NM

30.

1622

.74

30.1

162

2.07

626.

2694

.80

1.01

043

6.90

6.27

436.

9043

9.71

94.8

81.

009

NM

30.

2623

.71

29.8

962

3.06

626.

9894

.82

1.01

143

7.59

6.57

437.

5844

0.27

94.8

01.

010

NM

40.

1627

.16

30.2

262

6.49

629.

7994

.72

1.01

543

9.31

8.05

439.

2844

2.36

94.7

61.

015

NM

40.

2629

.37

29.7

862

8.73

631.

7094

.52

1.01

844

0.89

8.06

440.

8644

3.75

94.8

21.

018

NM

50.

1629

.12

31.1

762

8.41

633.

6994

.62

1.02

243

9.97

6.80

439.

9644

4.84

94.8

41.

021

NM

50.

2634

.66

30.2

863

4.00

637.

8494

.50

1.02

844

2.72

7.32

442.

7044

7.92

94.7

41.

028

NM

60.

1634

.40

32.2

963

3.64

637.

5794

.52

1.02

844

4.82

9.65

444.

7644

7.74

94.7

61.

027

NM

60.

2643

.49

31.3

064

2.79

645.

6794

.50

1.04

144

9.61

8.86

449.

5645

3.46

94.7

61.

040

NM

70.

1637

.19

36.9

263

6.18

641.

7394

.90

1.03

444

6.21

10.1

844

6.14

450.

1794

.82

1.03

3N

M70.

2654

.31

40.1

865

3.14

656.

8594

.76

1.05

945

8.13

11.7

345

8.03

460.

6094

.80

1.05

7N

M80.

1638

.82

32.1

563

8.07

643.

8794

.68

1.03

845

0.65

11.6

245

0.54

452.

3894

.50

1.03

8N

M80.

2664

.63

30.7

766

3.98

667.

5794

.56

1.07

646

6.23

13.0

346

6.09

469.

3994

.48

1.07

7N

M90.

1641

.44

36.4

264

0.47

647.

0294

.50

1.04

344

9.81

10.9

844

9.73

454.

0094

.76

1.04

2N

M90.

2680

.84

38.2

367

9.83

683.

7194

.58

1.10

247

4.83

11.9

047

4.73

479.

6794

.82

1.10

1

120

Tab

leE

61:

Infe

ren

cefo

rth

e84

thp

erce

nti

leeµ

+σ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103

×10

3×

103

%L

en.

CD

332.2

1-6

.45

332.

1833

1.10

93.2

21.

000

235.

75-1

.74

235.

7723

4.69

94.3

41.

000

MI.

C90

363.8

40.

3836

3.87

336.

5491

.72

1.01

625

5.65

1.46

255.

6723

6.82

92.6

81.

009

MI.

C80

389.0

540

.02

387.

0334

8.58

92.0

01.

053

268.

2619

.80

267.

5624

1.31

92.4

81.

028

MI.

D90

2726

.01

50.9

427

25.8

168

3.43

93.8

22.

064

238.

706.

6923

8.63

237.

9794

.56

1.01

4M

I.D

80

344.8

720

.84

344.

2734

4.68

93.8

01.

041

240.

2210

.70

240.

0023

9.94

94.9

41.

022

CE

NS

355.5

93.

2535

5.62

354.

8893

.50

1.07

225

3.22

1.43

253.

2424

9.98

94.5

01.

065

NM

10.

133

2.4

4-6

.03

332.

4233

1.46

93.2

61.

001

235.

81-1

.68

235.

8223

4.89

94.4

01.

001

NM

10.

233

2.4

3-6

.03

332.

4133

1.47

93.2

61.

001

235.

79-1

.69

235.

8123

4.90

94.3

81.

001

NM

20.

133

3.7

8-6

.01

333.

7633

2.20

93.0

61.

003

236.

45-1

.75

236.

4723

5.40

94.4

41.

003

NM

20.

233

3.8

9-5

.87

333.

8733

2.32

93.1

21.

004

236.

46-1

.84

236.

4823

5.45

94.4

01.

003

NM

30.

133

4.6

0-6

.56

334.

5633

3.18

93.1

61.

006

236.

93-2

.05

236.

9523

6.12

94.5

41.

006

NM

30.

233

4.9

3-6

.66

334.

8933

3.50

93.1

21.

007

237.

02-2

.14

237.

0323

6.34

94.5

21.

007

NM

40.

133

4.7

7-6

.36

334.

7433

4.52

93.0

81.

010

238.

43-0

.67

238.

4523

7.24

94.4

41.

011

NM

40.

233

5.1

4-6

.67

335.

1133

5.34

93.0

61.

013

239.

11-0

.76

239.

1323

7.85

94.3

61.

013

NM

50.

133

6.2

8-5

.94

336.

2633

5.99

93.4

01.

015

239.

73-0

.75

239.

7623

8.20

94.2

21.

015

NM

50.

233

8.1

5-6

.06

338.

1333

7.88

93.1

21.

021

241.

11-0

.48

241.

1323

9.59

94.3

41.

021

NM

60.

133

9.4

0-3

.26

339.

4233

7.96

93.2

21.

021

240.

27-1

.08

240.

2923

9.11

94.4

01.

019

NM

60.

234

3.2

0-3

.02

343.

2234

1.72

93.1

21.

032

242.

75-1

.30

242.

7824

1.71

94.2

81.

030

NM

70.

133

9.1

5-4

.09

339.

1633

9.04

93.3

01.

024

241.

15-1

.51

241.

1723

9.93

94.3

01.

022

NM

70.

234

5.7

7-4

.43

345.

7734

5.64

93.3

21.

044

245.

26-0

.80

245.

2924

4.74

94.4

21.

043

NM

80.

134

0.1

0-3

.42

340.

1234

0.32

93.3

21.

028

242.

18-1

.04

242.

2124

0.77

94.3

81.

026

NM

80.

235

1.8

4-2

.47

351.

8735

1.69

93.1

01.

062

249.

360.

1024

9.39

248.

8794

.34

1.06

0N

M90.

133

9.9

9-4

.19

340.

0034

1.08

93.1

21.

030

243.

56-1

.09

243.

5824

1.41

94.5

01.

029

NM

90.

235

4.9

2-3

.67

354.

9335

8.96

93.5

01.

084

256.

841.

1925

6.86

254.

4294

.28

1.08

4

121

Tab

leE

62:

Infe

ren

cefo

rth

e84

thp

erce

nti

leeµ

+σ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

103

.55

0.40

103.

5610

5.29

95.3

81.

000

73.9

1-0

.20

73.9

174

.41

95.2

01.

000

MI.

C90

111

.59

1.36

111.

5910

5.73

93.2

01.

004

79.6

80.

6179

.68

74.6

893

.34

1.00

4M

I.C

80

114

.66

5.06

114.

5610

6.47

93.2

01.

011

81.8

42.

5281

.81

75.0

992

.70

1.00

9M

I.D

90

104

.01

1.67

104.

0010

5.83

95.3

61.

005

74.3

10.

5874

.31

74.7

395

.24

1.00

4M

I.D

80

104

.83

2.83

104.

8010

6.31

95.3

21.

010

74.3

20.

9774

.32

75.0

195

.12

1.00

8C

EN

S110

.31

2.11

110.

3011

1.88

95.2

41.

063

78.5

40.

8978

.54

79.0

194

.92

1.06

2N

M10.

1103

.62

0.38

103.

6310

5.37

95.2

41.

001

73.9

6-0

.27

73.9

774

.46

95.2

01.

001

NM

10.

2103

.63

0.36

103.

6410

5.37

95.3

01.

001

73.9

7-0

.27

73.9

874

.47

95.1

41.

001

NM

20.

1103

.87

0.40

103.

8810

5.61

95.2

81.

003

74.2

2-0

.14

74.2

374

.63

95.1

61.

003

NM

20.

2103

.88

0.33

103.

8910

5.63

95.2

21.

003

74.2

3-0

.15

74.2

474

.65

95.1

41.

003

NM

30.

1104

.17

0.63

104.

1810

5.96

95.3

81.

006

74.5

1-0

.35

74.5

274

.87

95.1

41.

006

NM

30.

2104

.25

0.59

104.

2610

6.06

95.4

01.

007

74.6

1-0

.31

74.6

274

.94

95.0

01.

007

NM

40.

1104

.70

0.58

104.

7110

6.37

95.2

61.

010

74.8

4-0

.12

74.8

575

.16

95.0

01.

010

NM

40.

2104

.97

0.47

104.

9810

6.63

95.3

21.

013

75.0

9-0

.13

75.0

975

.36

95.2

01.

013

NM

50.

1104

.89

0.69

104.

8910

6.81

95.3

41.

014

74.8

5-0

.32

74.8

675

.46

95.0

41.

014

NM

50.

2105

.54

0.46

105.

5510

7.40

95.2

41.

020

75.2

4-0

.26

75.2

475

.89

94.9

41.

020

NM

60.

1105

.60

0.78

105.

6110

7.24

95.2

01.

019

75.3

20.

0475

.32

75.7

895

.16

1.01

8N

M60.

2106

.80

0.48

106.

8110

8.39

95.3

01.

030

76.1

0-0

.13

76.1

176

.60

95.4

41.

029

NM

70.

1105

.70

1.39

105.

7010

7.67

95.5

41.

023

75.5

00.

1175

.51

76.0

694

.94

1.02

2N

M70.

2107

.91

1.64

107.

9010

9.79

95.4

01.

043

77.1

30.

2377

.14

77.5

594

.90

1.04

2N

M80.

1106

.07

0.68

106.

0810

7.96

95.2

21.

025

76.2

30.

2776

.23

76.3

094

.94

1.02

5N

M80.

2109

.89

0.07

109.

9011

1.47

95.5

01.

059

78.4

90.

3178

.49

78.8

295

.24

1.05

9N

M90.

1106

.45

1.26

106.

4510

8.28

95.3

81.

028

75.8

60.

1875

.87

76.4

995

.24

1.02

8N

M90.

2112

.57

0.97

112.

5811

3.95

95.1

61.

082

79.9

10.

0679

.92

80.5

195

.18

1.08

2

122

Tab

leE

63:

Infe

ren

cefo

rth

e95

thp

erce

nti

leeµ

+1.6

45σ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

795.1

0-1

7.31

794.

9979

2.17

92.8

01.

000

558.

99-8

.76

558.

9756

0.54

93.8

61.

000

MI.

C90

894.7

112

.72

894.

7181

3.36

91.1

21.

027

621.

425.

1862

1.46

568.

4892

.14

1.01

4M

I.C

8095

8.3

712

0.03

950.

9285

2.39

91.8

21.

076

653.

2155

.31

650.

9358

2.69

92.3

21.

040

MI.

D90

202181

2.4

22864

0.33

2021

811.

7432

0046

.09

93.8

240

4.01

156

9.14

19.7

856

8.85

571.

7594

.40

1.02

0M

I.D

80

839.5

871

.98

836.

5783

8.69

93.8

61.

059

574.

0232

.10

573.

1857

7.74

94.7

41.

031

CE

NS

868.6

811

.97

868.

6886

7.27

92.8

21.

095

614.

641.

3961

4.70

608.

4894

.18

1.08

6N

M10.1

795.9

5-1

6.09

795.

8779

3.28

92.9

01.

001

559.

25-8

.57

559.

2456

1.16

93.8

01.

001

NM

10.2

795.9

5-1

6.10

795.

8779

3.31

92.9

01.

001

559.

22-8

.61

559.

2156

1.17

93.7

81.

001

NM

20.1

800.1

6-1

5.93

800.

0879

5.61

92.6

01.

004

561.

20-8

.76

561.

1956

2.73

93.7

81.

004

NM

20.2

800.4

7-1

5.56

800.

4079

5.94

92.6

61.

005

561.

25-8

.98

561.

2456

2.87

93.7

41.

004

NM

30.1

802.5

2-1

7.24

802.

4279

8.68

92.4

81.

008

562.

50-9

.51

562.

4756

4.96

93.8

21.

008

NM

30.2

803.6

0-1

7.45

803.

4979

9.57

92.5

41.

009

562.

97-9

.73

562.

9456

5.59

93.9

21.

009

NM

40.1

803.1

8-1

6.61

803.

0980

2.84

92.5

01.

013

567.

50-5

.63

567.

5356

8.44

93.7

41.

014

NM

40.2

804.5

2-1

7.32

804.

4180

5.13

92.6

21.

016

569.

46-5

.81

569.

4857

0.15

93.4

81.

017

NM

50.1

806.5

2-1

5.56

806.

4580

7.39

92.6

81.

019

571.

16-5

.83

571.

1957

1.44

94.2

81.

019

NM

50.2

812.1

3-1

5.67

812.

0681

2.71

92.6

01.

026

574.

89-5

.06

574.

9357

5.33

94.0

61.

026

NM

60.1

818.3

3-7

.68

818.

3881

3.65

92.8

01.

027

573.

57-6

.55

573.

5957

4.31

93.9

81.

025

NM

60.2

829.0

2-6

.75

829.

0782

4.12

92.5

81.

040

580.

69-6

.96

580.

7158

1.52

93.7

41.

037

NM

70.1

817.3

8-9

.97

817.

4181

7.05

92.8

61.

031

575.

90-7

.72

575.

9057

6.88

93.9

61.

029

NM

70.2

837.0

4-1

0.09

837.

0683

5.26

92.7

61.

054

587.

26-5

.68

587.

2959

0.08

94.1

21.

053

NM

80.1

820.5

9-7

.94

820.

6382

1.07

92.7

21.

036

578.

54-6

.42

578.

5657

9.48

93.9

81.

034

NM

80.2

853.3

8-4

.51

853.

4585

2.00

92.5

81.

076

598.

49-3

.09

598.

5460

1.41

93.9

41.

073

NM

90.1

818.9

8-1

0.23

818.

9982

3.42

92.9

41.

039

583.

83-6

.36

583.

8558

1.54

94.2

21.

037

NM

90.2

858.7

4-7

.94

858.

7987

0.43

93.0

01.

099

619.

14-0

.02

619.

2161

5.71

93.8

41.

098

123

Tab

leE

64:

Infe

ren

cefo

rth

e95

thp

erce

nti

leeµ

+1.6

45σ

bas

edon

LN

(µ=

0,σ

2=

1)d

ata

wit

hC

=3.

602

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

248.

101.

1024

8.12

251.

4495

.10

1.00

017

5.79

-1.4

817

5.80

177.

6295

.22

1.00

0M

I.C

90272.

414.

7527

2.40

252.

9192

.58

1.00

619

2.96

1.25

192.

9717

8.49

92.4

41.

005

MI.

C80

279.

7414

.88

279.

3725

5.10

92.7

41.

015

197.

796.

4319

7.71

179.

6792

.42

1.01

2M

I.D

90249.

495.

5824

9.45

253.

1595

.06

1.00

717

6.98

1.13

176.

9917

8.60

95.3

41.

006

MI.

D80

251.

789.

0025

1.64

254.

4495

.10

1.01

217

7.04

2.35

177.

0417

9.33

95.3

01.

010

CE

NS

268.

916.

0326

8.87

271.

9595

.12

1.08

219

0.25

1.67

190.

2619

1.93

94.9

21.

081

NM

10.1

248.

311.

0324

8.33

251.

7095

.12

1.00

117

5.95

-1.6

517

5.96

177.

7995

.30

1.00

1N

M10

.2248.

351.

0024

8.37

251.

7095

.16

1.00

117

5.96

-1.6

617

5.97

177.

8095

.28

1.00

1N

M20

.1249.

101.

1124

9.12

252.

4295

.08

1.00

417

6.75

-1.2

817

6.76

178.

3195

.18

1.00

4N

M20

.2249.

180.

9324

9.20

252.

4895

.06

1.00

417

6.78

-1.3

217

6.80

178.

3795

.20

1.00

4N

M30

.1249.

961.

7324

9.98

253.

5195

.36

1.00

817

7.65

-1.8

617

7.66

179.

0395

.06

1.00

8N

M30

.2250.

241.

6225

0.26

253.

8095

.42

1.00

917

7.91

-1.7

517

7.92

179.

2495

.02

1.00

9N

M40

.1251.

581.

6225

1.60

254.

7795

.06

1.01

317

8.57

-1.2

217

8.58

179.

9595

.22

1.01

3N

M40

.2252.

401.

3625

2.43

255.

5294

.90

1.01

617

9.19

-1.2

517

9.20

180.

5095

.08

1.01

6N

M50

.1252.

201.

9225

2.22

256.

1495

.08

1.01

917

8.78

-1.7

417

8.79

180.

8895

.02

1.01

8N

M50

.2254.

191.

3725

4.21

257.

7895

.06

1.02

517

9.85

-1.6

017

9.86

182.

0795

.22

1.02

5N

M60

.1254.

232.

1825

4.25

257.

4894

.86

1.02

418

0.28

-0.7

618

0.29

181.

8695

.24

1.02

4N

M60

.2257.

721.

4525

7.74

260.

6895

.20

1.03

718

2.27

-1.1

818

2.29

184.

1495

.18

1.03

7N

M70

.1254.

763.

8825

4.76

258.

8395

.24

1.02

918

0.82

-0.5

818

0.84

182.

7295

.20

1.02

9N

M70

.2261.

084.

5926

1.06

264.

6595

.12

1.05

318

5.35

-0.2

418

5.37

186.

8294

.94

1.05

2N

M80

.1255.

811.

9525

5.83

259.

7394

.96

1.03

318

2.73

-0.1

318

2.75

183.

4894

.72

1.03

3N

M80

.2266.

100.

4826

6.13

269.

2195

.12

1.07

118

8.88

0.04

188.

9019

0.27

95.0

61.

071

NM

90.1

256.

723.

5425

6.72

260.

7295

.00

1.03

718

2.10

-0.3

718

2.11

184.

0895

.16

1.03

6N

M90

.2272.

712.

9027

2.72

275.

5595

.04

1.09

619

2.49

-0.6

219

2.51

194.

5995

.30

1.09

6

124

Tab

leE

65:

Infe

ren

cefo

rth

em

eanµ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

645

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

101

.30

2.27

101.

2899

.19

94.2

81.

000

70.2

60.

5870

.26

70.5

094

.70

1.00

0M

I.C

90

102

.12

0.47

102.

1398

.96

94.0

80.

998

70.7

6-0

.39

70.7

770

.42

94.6

80.

999

MI.

C80

103

.55

2.77

103.

5299

.41

93.8

21.

002

71.5

30.

8471

.53

70.6

194

.28

1.00

2M

I.D

90

103

.09

2.05

103.

0810

6.03

94.5

41.

069

70.3

00.

5970

.30

70.6

794

.80

1.00

2M

I.D

80

101

.40

2.22

101.

3899

.82

94.4

81.

006

70.2

80.

5670

.29

70.7

294

.70

1.00

3C

EN

S102

.05

3.02

102.

0199

.75

94.1

81.

006

70.4

00.

9370

.40

70.8

394

.66

1.00

5N

M10.

1101

.35

2.29

101.

3399

.23

94.3

21.

000

70.2

50.

6270

.26

70.5

394

.62

1.00

0N

M10.

2101

.33

2.29

101.

3299

.23

94.3

81.

000

70.2

50.

6270

.26

70.5

394

.62

1.00

0N

M20.

1101

.48

2.38

101.

4699

.31

94.4

01.

001

70.3

20.

7470

.32

70.6

094

.64

1.00

1N

M20.

2101

.41

2.40

101.

4099

.33

94.4

01.

001

70.3

70.

7270

.37

70.6

194

.62

1.00

2N

M30.

1101

.61

2.66

101.

5899

.44

94.3

61.

002

70.3

40.

6970

.34

70.6

394

.78

1.00

2N

M30.

2101

.54

2.66

101.

5299

.51

94.5

21.

003

70.4

50.

6670

.46

70.6

894

.84

1.00

2N

M40.

1101

.66

2.63

101.

6499

.48

94.3

01.

003

70.3

10.

7770

.31

70.6

894

.92

1.00

3N

M40.

2101

.70

2.55

101.

6899

.63

94.3

61.

004

70.5

30.

8170

.53

70.8

095

.06

1.00

4N

M50.

1101

.72

2.74

101.

7099

.54

94.4

21.

003

70.3

30.

7870

.33

70.7

094

.78

1.00

3N

M50.

2101

.76

2.74

101.

7399

.85

94.2

61.

007

70.6

20.

8570

.63

70.9

394

.68

1.00

6N

M60.

1101

.75

2.73

101.

7399

.55

94.3

61.

004

70.2

60.

7370

.26

70.7

194

.86

1.00

3N

M60.

2101

.91

2.70

101.

8910

0.08

94.3

41.

009

70.7

70.

7770

.78

71.0

894

.82

1.00

8N

M70.

1101

.74

2.76

101.

7199

.57

94.2

41.

004

70.3

00.

8170

.30

70.7

394

.70

1.00

3N

M70.

2101

.96

2.77

101.

9310

0.38

94.4

41.

012

70.8

20.

9570

.83

71.3

294

.60

1.01

2N

M80.

1101

.84

2.74

101.

8199

.58

94.4

01.

004

70.3

30.

8670

.33

70.7

594

.76

1.00

3N

M80.

2102

.65

2.93

102.

6210

0.76

94.5

01.

016

71.3

51.

0571

.35

71.5

894

.60

1.01

5N

M90.

1101

.88

2.86

101.

8599

.61

94.3

61.

004

70.4

10.

8370

.41

70.7

494

.76

1.00

3N

M90.

2102

.80

3.03

102.

7610

1.19

94.3

61.

020

71.8

61.

1171

.86

71.8

794

.62

1.01

9

125

Tab

leE

66:

Infe

ren

cefo

rth

em

eanµ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

645

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

31.6

00.

0831

.61

31.6

194

.86

1.00

021

.92

-0.1

421

.92

22.3

595

.44

1.00

0M

I.C

90

31.7

8-0

.09

31.7

931

.61

94.7

21.

000

22.0

4-0

.19

22.0

422

.36

95.3

41.

000

MI.

C80

32.1

40.

0732

.14

31.6

394

.40

1.00

122

.22

0.01

22.2

222

.37

94.8

41.

001

MI.

D90

31.6

10.

0731

.61

31.6

394

.86

1.00

121

.93

-0.1

321

.93

22.3

695

.42

1.00

1M

I.D

80

31.6

00.

0631

.60

31.6

594

.86

1.00

121

.93

-0.1

721

.93

22.3

895

.52

1.00

1C

EN

S31

.69

0.12

31.6

931

.73

94.8

61.

004

21.9

8-0

.07

21.9

922

.44

95.5

01.

004

NM

10.

131

.61

0.07

31.6

131

.62

95.0

01.

000

21.9

2-0

.12

21.9

222

.36

95.4

01.

000

NM

10.

231

.61

0.07

31.6

131

.62

95.0

01.

000

21.9

2-0

.12

21.9

222

.36

95.4

01.

000

NM

20.

131

.63

0.08

31.6

331

.64

94.8

61.

001

21.9

3-0

.16

21.9

322

.37

95.4

81.

001

NM

20.

231

.63

0.08

31.6

431

.64

94.8

21.

001

21.9

4-0

.16

21.9

422

.38

95.5

01.

001

NM

30.

131

.64

0.12

31.6

431

.66

94.7

61.

002

21.9

5-0

.10

21.9

522

.39

95.3

21.

002

NM

30.

231

.66

0.13

31.6

631

.68

94.7

21.

002

21.9

7-0

.10

21.9

722

.40

95.1

41.

002

NM

40.

131

.65

0.11

31.6

531

.67

94.9

81.

002

21.9

6-0

.12

21.9

622

.40

95.4

81.

002

NM

40.

231

.71

0.11

31.7

131

.73

94.9

81.

004

22.0

1-0

.13

22.0

122

.44

95.5

81.

004

NM

50.

131

.70

0.10

31.7

031

.68

94.8

61.

002

21.9

6-0

.09

21.9

622

.41

95.4

81.

002

NM

50.

231

.83

0.10

31.8

331

.78

94.7

01.

006

22.0

2-0

.06

22.0

222

.48

95.3

61.

006

NM

60.

131

.68

0.12

31.6

831

.69

94.8

21.

003

21.9

8-0

.11

21.9

922

.41

95.4

21.

003

NM

60.

231

.85

0.17

31.8

531

.86

94.9

41.

008

22.0

5-0

.08

22.0

522

.53

95.1

41.

008

NM

70.

131

.65

0.09

31.6

531

.69

94.9

01.

003

21.9

4-0

.11

21.9

422

.41

95.4

61.

003

NM

70.

231

.86

0.10

31.8

631

.95

95.0

21.

011

22.0

5-0

.09

22.0

522

.60

95.4

21.

011

NM

80.

131

.70

0.14

31.7

031

.70

94.7

81.

003

21.9

7-0

.06

21.9

822

.42

95.4

01.

003

NM

80.

232

.10

0.16

32.1

032

.07

94.8

81.

015

22.1

60.

0222

.16

22.6

895

.32

1.01

5N

M90.

131

.62

0.11

31.6

331

.70

94.8

41.

003

21.9

7-0

.09

21.9

722

.42

95.5

61.

003

NM

90.

232

.06

0.11

32.0

732

.19

95.1

81.

018

22.2

4-0

.02

22.2

422

.77

95.3

81.

019

126

Tab

leE

67:

Infe

ren

cefo

rth

eva

rian

ceσ

2b

ased

onN

(µ=

0,σ

2=

1)d

ata

wit

hC

=1.

645

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

139.9

8-1

1.13

139.

5513

9.85

93.4

81.

000

98.8

5-3

.48

98.8

099

.65

94.5

01.

000

MI.

C90

150.4

3-1

5.42

149.

6513

9.76

91.5

00.

999

105.

98-5

.82

105.

8399

.68

92.7

21.

000

MI.

C80

155.3

1-7

.17

155.

1614

1.46

91.1

81.

012

109.

23-1

.21

109.

2310

0.41

92.5

21.

008

MI.

D90

510

50.1

716

31.1

651

029.

2124

02.4

394

.90

17.1

7999

.49

0.42

99.5

010

0.43

94.6

81.

008

MI.

D80

146.4

4-0

.72

146.

4514

3.31

94.2

81.

025

99.8

60.

7599

.87

100.

6794

.76

1.01

0C

EN

S14

6.8

7-7

.60

146.

6914

6.37

93.1

81.

047

103.

40-1

.71

103.

4010

4.01

94.4

61.

044

NM

10.1

140.2

6-1

1.02

139.

8414

0.31

93.4

21.

003

99.4

0-3

.25

99.3

510

0.00

94.5

61.

003

NM

10.2

140.2

7-1

1.01

139.

8514

0.33

93.4

21.

003

99.3

7-3

.26

99.3

310

0.01

94.5

21.

004

NM

20.1

141.6

8-1

0.65

141.

3014

1.36

93.1

41.

011

99.9

9-2

.73

99.9

710

0.76

94.5

21.

011

NM

20.2

141.9

2-1

0.57

141.

5414

1.51

93.1

21.

012

100.

03-2

.78

100.

0010

0.86

94.4

61.

012

NM

30.1

142.8

0-9

.31

142.

5114

2.54

93.3

61.

019

100.

48-2

.94

100.

4410

1.43

94.5

21.

018

NM

30.2

143.2

4-9

.27

142.

9514

3.05

93.4

01.

023

100.

85-3

.03

100.

8210

1.78

94.6

21.

021

NM

40.1

143.9

5-9

.41

143.

6514

3.22

93.3

61.

024

100.

93-2

.56

100.

9110

1.96

94.3

21.

023

NM

40.2

145.3

1-9

.67

145.

0114

4.40

93.3

61.

033

101.

79-2

.46

101.

7710

2.84

94.2

41.

032

NM

50.1

144.3

2-8

.89

144.

0614

3.77

93.2

61.

028

101.

71-2

.49

101.

6910

2.29

94.4

81.

026

NM

50.2

146.6

9-8

.88

146.

4314

6.07

93.0

61.

044

103.

53-2

.23

103.

5110

3.95

94.0

61.

043

NM

60.1

144.8

6-8

.93

144.

6014

4.08

93.3

41.

030

102.

10-2

.70

102.

0810

2.48

94.3

21.

028

NM

60.2

148.8

1-9

.04

148.

5514

7.71

93.1

01.

056

104.

99-2

.68

104.

9710

5.09

94.2

21.

055

NM

70.1

144.8

6-8

.84

144.

6014

4.31

93.2

01.

032

101.

97-2

.33

101.

9510

2.67

94.5

61.

030

NM

70.2

150.7

3-8

.70

150.

5014

9.52

92.8

61.

069

105.

81-1

.94

105.

8010

6.42

94.2

41.

068

NM

80.1

144.7

4-8

.91

144.

4814

4.47

93.2

21.

033

102.

05-2

.08

102.

0410

2.81

94.2

41.

032

NM

80.2

151.2

7-8

.32

151.

0615

1.34

93.2

81.

082

107.

12-1

.54

107.

1210

7.70

94.0

81.

081

NM

90.1

145.0

7-8

.38

144.

8414

4.69

92.9

41.

035

102.

21-2

.25

102.

1910

2.88

94.6

81.

032

NM

90.2

153.4

7-7

.89

153.

2815

3.06

93.2

21.

094

108.

08-1

.67

108.

0810

8.86

94.1

41.

092

127

Tab

leE

68:

Infe

ren

cefo

rth

eva

rian

ceσ

2b

ased

onN

(µ=

0,σ

2=

1)d

ata

wit

hC

=1.

645

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

44.8

3-0

.51

44.8

344

.70

95.0

01.

000

31.6

4-0

.53

31.6

431

.61

94.8

81.

000

MI.

C90

47.6

1-0

.92

47.6

144

.76

93.0

21.

001

33.7

2-0

.56

33.7

231

.66

93.8

41.

002

MI.

C80

48.8

0-0

.23

48.8

144

.88

92.9

01.

004

34.8

60.

1734

.86

31.7

492

.84

1.00

4M

I.D

90

44.9

90.

1445

.00

44.8

494

.84

1.00

331

.64

-0.1

831

.64

31.6

994

.98

1.00

3M

I.D

80

44.9

70.

2544

.98

44.9

195

.06

1.00

531

.82

-0.2

331

.82

31.7

494

.94

1.00

4C

EN

S46

.45

-0.3

246

.45

46.5

494

.96

1.04

133

.06

-0.2

133

.07

32.9

195

.12

1.04

1N

M10.

144

.88

-0.5

544

.88

44.8

494

.78

1.00

331

.72

-0.4

731

.72

31.7

194

.92

1.00

3N

M10.

244

.90

-0.5

644

.90

44.8

594

.78

1.00

331

.73

-0.4

731

.73

31.7

294

.84

1.00

3N

M20.

145

.14

-0.5

245

.14

45.1

694

.90

1.01

031

.92

-0.6

031

.92

31.9

394

.98

1.01

0N

M20.

245

.25

-0.5

145

.25

45.2

194

.90

1.01

131

.97

-0.6

331

.97

31.9

695

.00

1.01

1N

M30.

145

.64

-0.3

345

.64

45.4

894

.58

1.01

732

.15

-0.3

632

.15

32.1

694

.92

1.01

7N

M30.

245

.91

-0.2

645

.91

45.6

494

.62

1.02

132

.27

-0.3

732

.27

32.2

794

.92

1.02

1N

M40.

145

.77

-0.3

645

.78

45.6

994

.74

1.02

232

.30

-0.4

432

.30

32.3

095

.16

1.02

2N

M40.

246

.29

-0.3

646

.29

46.0

894

.82

1.03

132

.57

-0.4

732

.57

32.5

895

.32

1.03

1N

M50.

145

.78

-0.4

145

.78

45.8

395

.06

1.02

532

.47

-0.3

232

.47

32.4

194

.98

1.02

5N

M50.

246

.61

-0.4

046

.61

46.5

694

.82

1.04

232

.98

-0.2

032

.98

32.9

395

.08

1.04

2N

M60.

145

.87

-0.2

945

.87

45.9

394

.82

1.02

732

.55

-0.3

832

.55

32.4

795

.02

1.02

7N

M60.

247

.08

-0.1

447

.08

47.1

194

.72

1.05

433

.57

-0.2

933

.57

33.3

095

.12

1.05

4N

M70.

146

.02

-0.4

546

.02

45.9

894

.82

1.02

932

.53

-0.3

932

.53

32.5

195

.02

1.02

9N

M70.

247

.96

-0.4

147

.96

47.6

594

.68

1.06

633

.72

-0.3

333

.72

33.6

995

.10

1.06

6N

M80.

146

.13

-0.2

346

.13

46.0

494

.78

1.03

032

.72

-0.1

832

.72

32.5

695

.14

1.03

0N

M80.

248

.59

-0.1

048

.60

48.2

294

.96

1.07

934

.46

0.05

34.4

634

.10

94.8

61.

079

NM

90.

146

.00

-0.3

446

.01

46.0

894

.70

1.03

132

.67

-0.3

132

.67

32.5

895

.06

1.03

1N

M90.

248

.97

-0.3

748

.97

48.7

395

.02

1.09

034

.65

-0.1

034

.65

34.4

794

.86

1.09

0

128

Tab

leE

69:

Infe

ren

cefo

rth

e84

thp

erce

nti

leµ

+σ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

645

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

121.9

9-5

.79

121.

8612

1.49

94.4

61.

000

84.8

4-2

.39

84.8

286

.35

95.8

21.

000

MI.

C90

128.8

8-1

0.58

128.

4612

1.34

92.8

40.

999

89.2

8-4

.93

89.1

686

.34

94.3

21.

000

MI.

C80

133.1

9-4

.76

133.

1212

2.05

92.3

01.

005

92.0

6-1

.69

92.0

586

.68

93.8

01.

004

MI.

D90

223.9

110

.47

223.

6913

4.19

95.2

61.

105

85.0

8-0

.75

85.0

886

.69

95.6

81.

004

MI.

D80

122.2

5-1

.98

122.

2512

2.65

94.9

61.

010

85.2

0-0

.79

85.2

186

.85

95.8

01.

006

CE

NS

126.6

1-3

.51

126.

5712

5.20

94.4

41.

031

87.0

7-1

.26

87.0

788

.86

95.6

01.

029

NM

10.1

122.2

3-5

.73

122.

1112

1.76

94.6

01.

002

85.0

6-2

.25

85.0

486

.55

95.6

41.

002

NM

10.2

122.2

0-5

.71

122.

0812

1.77

94.6

01.

002

85.0

5-2

.25

85.0

386

.56

95.6

61.

002

NM

20.1

123.1

8-5

.50

123.

0712

2.36

94.3

61.

007

85.4

6-1

.89

85.4

586

.99

95.4

01.

007

NM

20.2

123.1

4-5

.45

123.

0312

2.48

94.4

81.

008

85.5

9-1

.93

85.5

887

.06

95.2

81.

008

NM

30.1

123.9

0-4

.59

123.

8312

3.03

94.3

61.

013

85.7

6-2

.05

85.7

487

.38

95.6

81.

012

NM

30.2

123.9

6-4

.57

123.

8912

3.41

94.4

01.

016

86.1

8-2

.13

86.1

687

.65

95.7

21.

015

NM

40.1

124.4

7-4

.70

124.

3912

3.42

94.3

81.

016

85.8

4-1

.79

85.8

387

.69

95.7

61.

016

NM

40.2

125.1

9-4

.96

125.

1112

4.32

94.3

81.

023

86.7

0-1

.72

86.6

988

.34

95.5

21.

023

NM

50.1

124.7

8-4

.34

124.

7112

3.73

94.4

21.

018

86.2

1-1

.76

86.2

187

.87

95.4

81.

018

NM

50.2

125.9

3-4

.43

125.

8712

5.48

94.4

61.

033

87.7

2-1

.61

87.7

189

.13

95.5

01.

032

NM

60.1

125.0

8-4

.40

125.

0212

3.91

94.4

01.

020

86.1

8-1

.93

86.1

687

.99

95.5

01.

019

NM

60.2

127.3

1-4

.62

127.

2412

6.73

94.3

21.

043

88.6

3-1

.96

88.6

290

.01

95.2

21.

042

NM

70.1

125.0

6-4

.32

125.

0012

4.04

94.5

21.

021

86.2

4-1

.66

86.2

488

.10

95.3

41.

020

NM

70.2

128.3

7-4

.46

128.

3012

8.16

94.4

61.

055

89.1

6-1

.42

89.1

591

.05

95.0

41.

054

NM

80.1

125.2

3-4

.37

125.

1712

4.14

94.4

01.

022

86.3

5-1

.49

86.3

588

.17

95.5

81.

021

NM

80.2

129.8

8-4

.12

129.

8212

9.71

94.5

21.

068

90.7

8-1

.15

90.7

992

.14

95.2

41.

067

NM

90.1

125.4

6-4

.00

125.

4112

4.25

94.4

41.

023

86.6

2-1

.61

86.6

288

.22

95.6

01.

022

NM

90.2

131.2

1-3

.89

131.

1613

1.30

94.4

61.

081

92.2

6-1

.19

92.2

693

.25

95.3

01.

080

129

Tab

leE

70:

Infe

ren

cefo

rth

e84

thp

erce

nti

leµ

+σ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

645

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

38.7

7-0

.43

38.7

738

.71

94.5

81.

000

27.0

8-0

.53

27.0

727

.38

95.4

61.

000

MI.

C90

40.4

1-0

.87

40.4

038

.75

93.7

61.

001

28.2

8-0

.63

28.2

727

.41

94.5

41.

001

MI.

C80

41.6

5-0

.42

41.6

538

.82

92.9

61.

003

29.1

7-0

.10

29.1

827

.46

93.6

61.

003

MI.

D90

38.8

5-0

.17

38.8

538

.79

94.5

21.

002

27.0

9-0

.37

27.0

927

.43

95.3

41.

002

MI.

D80

38.8

1-0

.16

38.8

138

.86

94.6

61.

004

27.1

8-0

.45

27.1

827

.47

95.4

41.

004

CE

NS

39.6

5-0

.31

39.6

639

.79

94.7

81.

028

27.8

4-0

.31

27.8

428

.14

95.5

01.

028

NM

10.

138

.81

-0.4

638

.81

38.8

094

.68

1.00

227

.12

-0.4

827

.12

27.4

495

.54

1.00

2N

M10.

238

.82

-0.4

738

.82

38.8

094

.58

1.00

227

.12

-0.4

827

.12

27.4

495

.46

1.00

2N

M20.

138

.97

-0.4

438

.98

38.9

894

.90

1.00

727

.22

-0.5

927

.22

27.5

795

.42

1.00

7N

M20.

239

.03

-0.4

339

.04

39.0

294

.90

1.00

827

.27

-0.6

027

.26

27.5

995

.58

1.00

8N

M30.

139

.19

-0.3

139

.19

39.1

794

.44

1.01

227

.37

-0.4

127

.37

27.7

095

.54

1.01

2N

M30.

239

.36

-0.2

639

.36

39.2

994

.32

1.01

527

.46

-0.4

227

.46

27.7

895

.66

1.01

5N

M40.

139

.28

-0.3

439

.28

39.2

994

.78

1.01

527

.46

-0.4

727

.46

27.7

995

.52

1.01

5N

M40.

239

.63

-0.3

439

.63

39.5

994

.82

1.02

327

.68

-0.4

927

.68

27.9

995

.44

1.02

3N

M50.

139

.41

-0.3

739

.41

39.3

794

.62

1.01

727

.53

-0.3

927

.53

27.8

595

.54

1.01

7N

M50.

240

.07

-0.3

740

.07

39.9

394

.52

1.03

227

.89

-0.2

927

.89

28.2

495

.64

1.03

2N

M60.

139

.40

-0.2

839

.40

39.4

394

.72

1.01

927

.63

-0.4

327

.63

27.8

895

.60

1.01

9N

M60.

240

.31

-0.1

840

.31

40.3

494

.84

1.04

228

.26

-0.3

728

.26

28.5

395

.06

1.04

2N

M70.

139

.39

-0.4

039

.39

39.4

794

.78

1.02

027

.49

-0.4

327

.49

27.9

195

.80

1.02

0N

M70.

240

.74

-0.4

040

.74

40.7

894

.86

1.05

428

.29

-0.4

028

.29

28.8

495

.80

1.05

4N

M80.

139

.56

-0.2

539

.57

39.5

094

.74

1.02

027

.67

-0.2

927

.67

27.9

395

.66

1.02

0N

M80.

241

.53

-0.1

841

.54

41.2

794

.84

1.06

628

.89

-0.1

028

.89

29.1

995

.58

1.06

6N

M90.

139

.30

-0.3

339

.30

39.5

294

.98

1.02

127

.63

-0.3

827

.63

27.9

595

.50

1.02

1N

M90.

241

.60

-0.3

741

.60

41.7

794

.98

1.07

929

.14

-0.2

129

.14

29.5

495

.46

1.07

9

130

Tab

leE

71:

Infe

ren

cefo

rth

e95

thp

erce

nti

leµ

+1.6

45σ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

645

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

152.

16-1

0.99

151.

7815

2.16

94.5

01.

000

106.

15-4

.30

106.

0710

8.14

95.6

01.

000

MI.

C90

163.

50-1

7.70

162.

5515

2.04

92.5

00.

999

113.

53-7

.86

113.

2710

8.18

93.6

01.

000

MI.

C80

169.

45-9

.62

169.

2015

2.98

91.5

01.

005

117.

50-3

.33

117.

4610

8.65

93.0

01.

005

MI.

D90

347.

3515

.89

347.

0217

1.14

95.4

01.

125

106.

50-1

.62

106.

5010

8.64

95.6

01.

005

MI.

D80

152.

50-4

.68

152.

4415

3.76

94.8

01.

011

106.

75-1

.66

106.

7510

8.87

95.7

01.

007

CE

NS

159.

49-7

.71

159.

3215

8.28

94.4

01.

040

110.

08-2

.68

110.

0611

2.33

95.5

01.

039

NM

10.1

152.

53-1

0.90

152.

1615

2.61

94.5

01.

003

106.

56-4

.10

106.

4910

8.48

95.6

01.

003

NM

10.2

152.

49-1

0.88

152.

1215

2.64

94.5

01.

003

106.

54-4

.10

106.

4810

8.49

95.5

01.

003

NM

20.1

154.

08-1

0.58

153.

7315

3.62

94.4

01.

010

107.

20-3

.58

107.

1510

9.21

95.5

01.

010

NM

20.2

154.

11-1

0.52

153.

7715

3.79

94.3

01.

011

107.

35-3

.63

107.

3010

9.33

95.5

01.

011

NM

30.1

155.

20-9

.26

154.

9415

4.69

94.3

01.

017

107.

70-3

.82

107.

6510

9.87

95.6

01.

016

NM

30.2

155.

41-9

.24

155.

1515

5.29

94.6

01.

021

108.

30-3

.94

108.

2411

0.29

95.6

01.

020

NM

40.1

156.

18-9

.43

155.

9115

5.36

94.3

01.

021

107.

92-3

.44

107.

8811

0.38

95.8

01.

021

NM

40.2

157.

45-9

.81

157.

1615

6.76

94.2

01.

030

109.

16-3

.35

109.

1211

1.40

95.5

01.

030

NM

50.1

156.

63-8

.91

156.

4015

5.85

94.4

01.

024

108.

60-3

.40

108.

5611

0.69

95.5

01.

024

NM

50.2

158.

70-9

.05

158.

4615

8.57

94.4

01.

042

110.

88-3

.20

110.

8511

2.64

95.5

01.

042

NM

60.1

157.

15-8

.99

156.

9115

6.16

94.4

01.

026

108.

68-3

.64

108.

6311

0.89

95.3

01.

025

NM

60.2

160.

99-9

.35

160.

7416

0.50

94.0

01.

055

112.

38-3

.71

112.

3311

4.00

95.2

01.

054

NM

70.1

157.

15-8

.89

156.

9215

6.38

94.3

01.

028

108.

72-3

.26

108.

6811

1.06

95.5

01.

027

NM

70.2

162.

82-9

.12

162.

5816

2.67

94.3

01.

069

113.

23-2

.95

113.

2011

5.56

95.3

01.

069

NM

80.1

157.

30-8

.96

157.

0615

6.54

94.4

01.

029

108.

85-3

.00

108.

8211

1.19

95.4

01.

028

NM

80.2

164.

59-8

.67

164.

3816

4.96

94.5

01.

084

115.

44-2

.58

115.

4211

7.18

95.0

01.

084

NM

90.1

157.

65-8

.42

157.

4415

6.72

94.4

01.

030

109.

22-3

.18

109.

1811

1.27

95.5

01.

029

NM

90.2

166.

79-8

.35

166.

5916

7.24

94.4

01.

099

117.

37-2

.67

117.

3511

8.78

95.0

01.

098

131

Tab

leE

72:

Infe

ren

cefo

rth

e95

thp

erce

nti

leµ

+1.6

45σ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

645

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

48.6

0-0

.76

48.5

948

.48

94.7

01.

000

34.0

9-0

.78

34.0

834

.29

95.4

01.

000

MI.

C90

51.3

3-1

.38

51.3

248

.55

93.5

01.

001

36.1

1-0

.92

36.1

034

.34

94.0

01.

002

MI.

C80

53.0

8-0

.74

53.0

848

.66

92.9

01.

004

37.4

7-0

.17

37.4

734

.42

92.9

01.

004

MI.

D90

48.7

3-0

.32

48.7

348

.61

94.6

01.

003

34.1

0-0

.52

34.1

034

.37

95.4

01.

002

MI.

D80

48.6

7-0

.30

48.6

848

.71

94.6

01.

005

34.2

5-0

.64

34.2

534

.44

95.2

01.

004

CE

NS

50.1

0-0

.59

50.1

150

.30

95.3

01.

037

35.4

0-0

.46

35.4

035

.57

95.6

01.

037

NM

10.

148

.65

-0.8

048

.65

48.6

394

.80

1.00

334

.16

-0.7

234

.16

34.3

995

.40

1.00

3N

M10.

248

.67

-0.8

148

.67

48.6

394

.70

1.00

334

.17

-0.7

134

.16

34.4

095

.40

1.00

3N

M20.

148

.93

-0.7

748

.93

48.9

494

.80

1.01

034

.34

-0.8

634

.34

34.6

195

.40

1.00

9N

M20.

249

.03

-0.7

649

.03

49.0

094

.60

1.01

134

.41

-0.8

934

.40

34.6

595

.40

1.01

1N

M30.

149

.32

-0.5

849

.32

49.2

594

.70

1.01

634

.59

-0.6

134

.58

34.8

395

.80

1.01

6N

M30.

249

.60

-0.5

149

.60

49.4

494

.70

1.02

034

.72

-0.6

234

.72

34.9

695

.60

1.02

0N

M40.

149

.46

-0.6

249

.46

49.4

695

.00

1.02

034

.73

-0.6

934

.73

34.9

895

.50

1.02

0N

M40.

250

.03

-0.6

350

.03

49.9

294

.80

1.03

035

.07

-0.7

335

.06

35.3

095

.60

1.03

0N

M50.

149

.62

-0.6

749

.62

49.6

094

.60

1.02

334

.87

-0.5

734

.87

35.0

795

.60

1.02

3N

M50.

250

.63

-0.6

750

.63

50.4

694

.70

1.04

135

.44

-0.4

435

.44

35.6

995

.80

1.04

1N

M60.

149

.64

-0.5

549

.64

49.6

995

.00

1.02

535

.01

-0.6

435

.00

35.1

495

.20

1.02

5N

M60.

251

.04

-0.4

051

.05

51.1

095

.20

1.05

436

.04

-0.5

536

.04

36.1

395

.20

1.05

4N

M70.

149

.67

-0.7

249

.67

49.7

594

.80

1.02

634

.83

-0.6

534

.83

35.1

895

.70

1.02

6N

M70.

251

.82

-0.7

251

.82

51.7

694

.90

1.06

836

.12

-0.6

036

.12

36.6

195

.90

1.06

8N

M80.

149

.90

-0.4

949

.90

49.8

195

.10

1.02

735

.10

-0.4

335

.10

35.2

295

.40

1.02

7N

M80.

252

.90

-0.4

152

.90

52.4

995

.20

1.08

337

.02

-0.1

837

.03

37.1

295

.30

1.08

3N

M90.

149

.56

-0.6

149

.56

49.8

494

.90

1.02

835

.04

-0.5

735

.04

35.2

595

.60

1.02

8N

M90.

253

.09

-0.6

853

.09

53.2

095

.00

1.09

737

.36

-0.3

437

.37

37.6

395

.20

1.09

8

132

Tab

leE

73:

Infe

ren

cefo

rµ

+σ

2/2

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

645

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103×

103×

103

%L

en.

CD

121.6

9-3

.30

121.

6512

1.43

94.4

01.

000

84.7

7-1

.16

84.7

886

.35

95.8

01.

000

MI.

C90

128.5

6-7

.24

128.

3712

1.30

92.9

00.

999

89.1

9-3

.30

89.1

486

.35

94.3

01.

000

MI.

C80

133.3

4-0

.82

133.

3512

2.27

92.4

01.

007

92.1

30.

2392

.14

86.7

993

.80

1.00

5M

I.D

9025

524.

5581

7.63

2551

4.01

1252

.11

95.4

010

.311

85.1

40.

8185

.15

86.8

095

.70

1.00

5M

I.D

8012

3.4

71.

8612

3.47

123.

3395

.00

1.01

685

.30

0.94

85.3

086

.98

95.8

01.

007

CE

NS

126.5

4-0

.78

126.

5512

5.29

94.4

01.

032

87.0

60.

0887

.07

88.9

395

.60

1.03

0N

M10

.112

1.9

4-3

.22

121.

9112

1.71

94.6

01.

002

84.9

9-1

.01

84.9

986

.56

95.7

01.

002

NM

10.2

121.9

0-3

.21

121.

8712

1.72

94.7

01.

002

84.9

8-1

.01

84.9

986

.57

95.7

01.

002

NM

20.1

122.8

9-2

.94

122.

8712

2.33

94.4

01.

007

85.4

2-0

.63

85.4

287

.01

95.4

01.

008

NM

20.2

122.8

6-2

.89

122.

8312

2.44

94.5

01.

008

85.5

5-0

.67

85.5

687

.09

95.4

01.

008

NM

30.1

123.6

7-2

.00

123.

6612

3.03

94.5

01.

013

85.7

0-0

.78

85.7

187

.41

95.6

01.

012

NM

30.2

123.7

3-1

.97

123.

7312

3.41

94.5

01.

016

86.1

3-0

.85

86.1

487

.67

95.6

01.

015

NM

40.1

124.2

8-2

.07

124.

2812

3.43

94.5

01.

016

85.8

0-0

.51

85.8

087

.72

95.8

01.

016

NM

40.2

125.0

1-2

.29

125.

0012

4.32

94.5

01.

024

86.6

7-0

.42

86.6

888

.38

95.6

01.

023

NM

50.1

124.6

2-1

.70

124.

6312

3.75

94.4

01.

019

86.1

8-0

.47

86.1

987

.91

95.5

01.

018

NM

50.2

125.7

9-1

.70

125.

7912

5.50

94.6

01.

034

87.7

0-0

.26

87.7

189

.17

95.5

01.

033

NM

60.1

124.9

3-1

.74

124.

9312

3.94

94.4

01.

021

86.1

4-0

.62

86.1

588

.03

95.5

01.

019

NM

60.2

127.1

3-1

.82

127.

1312

6.77

94.3

01.

044

88.6

0-0

.57

88.6

190

.05

95.2

01.

043

NM

70.1

124.8

9-1

.66

124.

8912

4.08

94.5

01.

022

86.2

0-0

.36

86.2

088

.14

95.4

01.

021

NM

70.2

128.2

1-1

.58

128.

2212

8.22

94.5

01.

056

89.1

5-0

.02

89.1

691

.11

95.1

01.

055

NM

80.1

125.0

8-1

.71

125.

0812

4.18

94.5

01.

023

86.3

2-0

.18

86.3

388

.22

95.6

01.

022

NM

80.2

129.8

0-1

.23

129.

8012

9.79

94.7

01.

069

90.7

90.

2890

.79

92.2

195

.30

1.06

8N

M90

.112

5.3

4-1

.33

125.

3512

4.30

94.5

01.

024

86.6

0-0

.30

86.6

188

.27

95.6

01.

022

NM

90.2

131.1

9-0

.91

131.

2013

1.41

94.7

01.

082

92.2

90.

2792

.30

93.3

395

.40

1.08

1

133

Tab

leE

74:

Infe

ren

cefo

rµ

+σ

2/2

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

645

=95

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

38.7

7-0

.18

38.7

738

.71

94.5

01.

000

27.0

7-0

.40

27.0

727

.37

95.4

01.

000

MI.

C90

40.4

1-0

.55

40.4

138

.75

93.7

01.

001

28.2

7-0

.47

28.2

727

.41

94.6

01.

001

MI.

C80

41.6

7-0

.04

41.6

738

.83

93.0

01.

003

29.1

80.

1029

.18

27.4

793

.70

1.00

3M

I.D

90

38.8

60.

1438

.86

38.8

094

.50

1.00

227

.10

-0.2

227

.10

27.4

395

.40

1.00

2M

I.D

80

38.8

20.

1838

.83

38.8

794

.60

1.00

427

.18

-0.2

827

.18

27.4

895

.50

1.00

4C

EN

S39

.66

-0.0

439

.66

39.8

094

.70

1.02

827

.84

-0.1

727

.84

28.1

495

.50

1.02

8N

M10.

138

.80

-0.2

138

.81

38.8

094

.70

1.00

227

.12

-0.3

627

.12

27.4

495

.50

1.00

2N

M10.

238

.81

-0.2

138

.82

38.8

094

.60

1.00

227

.12

-0.3

527

.12

27.4

495

.50

1.00

2N

M20.

138

.97

-0.1

838

.97

38.9

994

.90

1.00

727

.22

-0.4

627

.22

27.5

795

.40

1.00

7N

M20.

239

.03

-0.1

739

.03

39.0

294

.90

1.00

827

.26

-0.4

827

.26

27.5

995

.60

1.00

8N

M30.

139

.19

-0.0

539

.20

39.1

794

.40

1.01

227

.37

-0.2

827

.37

27.7

095

.60

1.01

2N

M30.

239

.36

0.00

39.3

739

.30

94.3

01.

015

27.4

6-0

.29

27.4

627

.79

95.6

01.

015

NM

40.

139

.28

-0.0

739

.28

39.3

094

.80

1.01

527

.46

-0.3

427

.46

27.7

995

.60

1.01

5N

M40.

239

.64

-0.0

739

.64

39.5

994

.80

1.02

327

.68

-0.3

627

.68

27.9

995

.50

1.02

3N

M50.

139

.41

-0.1

039

.42

39.3

894

.70

1.01

727

.53

-0.2

527

.53

27.8

595

.50

1.01

7N

M50.

240

.08

-0.1

040

.08

39.9

494

.50

1.03

227

.89

-0.1

627

.89

28.2

495

.60

1.03

2N

M60.

139

.40

-0.0

239

.41

39.4

494

.70

1.01

927

.63

-0.3

027

.63

27.8

895

.60

1.01

9N

M60.

240

.31

0.10

40.3

240

.35

94.8

01.

042

28.2

6-0

.23

28.2

628

.53

95.0

01.

042

NM

70.

139

.39

-0.1

439

.39

39.4

794

.80

1.02

027

.49

-0.3

027

.49

27.9

195

.70

1.02

0N

M70.

240

.74

-0.1

140

.75

40.7

994

.80

1.05

428

.29

-0.2

628

.30

28.8

495

.80

1.05

4N

M80.

139

.57

0.02

39.5

739

.51

94.8

01.

020

27.6

7-0

.15

27.6

727

.94

95.7

01.

020

NM

80.

241

.54

0.11

41.5

441

.28

94.9

01.

066

28.9

00.

0428

.90

29.1

995

.60

1.06

6N

M90.

139

.30

-0.0

639

.31

39.5

395

.00

1.02

127

.63

-0.2

527

.63

27.9

595

.40

1.02

1N

M90.

241

.60

-0.0

741

.61

41.7

794

.90

1.07

929

.13

-0.0

629

.14

29.5

495

.50

1.07

9

134

Tab

leE

75:

Infe

ren

cefo

rth

em

eanµ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

282

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

98.0

1-1

.17

98.0

199

.14

94.7

21.

000

69.9

7-2

.11

69.9

570

.45

94.8

81.

000

MI.

C90

99.7

5-1

.86

99.7

499

.22

94.4

01.

001

71.4

7-2

.37

71.4

470

.50

94.4

41.

001

MI.

C80

102

.31

3.72

102.

2599

.95

93.7

81.

008

72.3

40.

0272

.35

70.7

994

.42

1.00

5M

I.D

90

98.2

1-1

.14

98.2

299

.74

94.8

81.

006

70.0

2-2

.13

70.0

070

.67

94.9

81.

003

MI.

D80

98.1

8-1

.28

98.1

999

.96

94.9

01.

008

70.0

1-2

.11

69.9

970

.83

94.8

81.

005

CE

NS

98.9

7-0

.26

98.9

810

0.35

94.9

81.

012

70.6

6-1

.76

70.6

471

.21

94.9

61.

011

NM

10.

198.1

3-1

.08

98.1

399

.21

94.5

81.

001

70.0

2-2

.07

70.0

070

.49

94.8

41.

001

NM

10.

298.1

2-1

.09

98.1

399

.21

94.5

61.

001

70.0

4-2

.07

70.0

170

.49

94.8

61.

001

NM

20.

198.1

2-0

.99

98.1

399

.35

94.9

81.

002

70.1

1-2

.09

70.0

870

.57

94.8

21.

002

NM

20.

298.1

5-0

.99

98.1

599

.37

94.9

01.

002

70.1

4-2

.11

70.1

270

.58

94.7

41.

002

NM

30.

198.1

9-0

.95

98.1

999

.50

94.8

61.

004

70.1

4-2

.03

70.1

270

.68

94.9

81.

003

NM

30.

298.1

5-1

.02

98.1

699

.56

94.7

81.

004

70.2

4-1

.95

70.2

270

.73

95.0

01.

004

NM

40.

198.4

0-0

.68

98.4

199

.68

94.7

01.

005

70.2

4-2

.05

70.2

270

.77

94.9

21.

005

NM

40.

298.5

7-0

.73

98.5

899

.84

94.6

21.

007

70.4

3-2

.05

70.4

170

.88

94.8

81.

006

NM

50.

198.6

8-0

.78

98.6

999

.75

94.7

81.

006

70.2

5-2

.19

70.2

270

.80

94.8

41.

005

NM

50.

299.1

1-0

.81

99.1

210

0.09

94.7

41.

010

70.5

0-2

.30

70.4

771

.04

95.0

41.

008

NM

60.

198.5

2-0

.75

98.5

399

.81

94.7

81.

007

70.4

2-1

.80

70.4

170

.91

94.9

41.

007

NM

60.

299.1

9-0

.97

99.1

910

0.42

95.0

01.

013

71.0

0-1

.65

70.9

971

.38

94.7

61.

013

NM

70.

198.8

2-0

.49

98.8

399

.91

94.8

41.

008

70.2

9-1

.96

70.2

770

.92

94.8

81.

007

NM

70.

299.4

2-0

.65

99.4

210

0.92

94.7

61.

018

71.0

3-1

.94

71.0

171

.66

94.9

01.

017

NM

80.

198.5

7-0

.55

98.5

899

.94

94.7

01.

008

70.4

3-1

.93

70.4

170

.95

94.9

61.

007

NM

80.

299.9

3-0

.64

99.9

410

1.45

94.7

41.

023

71.5

4-2

.03

71.5

272

.03

94.8

21.

022

NM

90.

198.4

9-0

.63

98.5

099

.95

94.8

81.

008

70.4

7-1

.84

70.4

570

.98

94.8

61.

008

NM

90.

299.9

2-0

.92

99.9

210

2.05

95.0

81.

029

72.3

2-1

.86

72.3

172

.50

94.6

81.

029

135

Tab

leE

76:

Infe

ren

cefo

rth

em

eanµ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

282

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

31.4

0-0

.27

31.4

031

.60

95.0

41.

000

22.2

9-0

.02

22.2

922

.34

95.2

01.

000

MI.

C90

31.8

7-0

.22

31.8

731

.63

94.6

21.

001

22.5

9-0

.11

22.5

922

.36

94.6

21.

001

MI.

C80

32.5

20.

2532

.52

31.6

994

.16

1.00

323

.01

0.20

23.0

122

.39

94.3

21.

002

MI.

D90

31.4

2-0

.24

31.4

331

.64

95.2

21.

001

22.3

1-0

.04

22.3

122

.37

94.9

81.

001

MI.

D80

31.5

1-0

.31

31.5

131

.70

95.0

81.

003

22.3

3-0

.02

22.3

422

.41

95.0

01.

003

CE

NS

31.6

9-0

.13

31.6

931

.93

95.0

01.

010

22.5

00.

0322

.50

22.5

794

.88

1.01

0N

M10.

131

.44

-0.2

931

.44

31.6

195

.02

1.00

022

.30

-0.0

322

.30

22.3

695

.10

1.00

0N

M10.

231

.44

-0.2

931

.44

31.6

295

.02

1.00

022

.30

-0.0

322

.30

22.3

695

.10

1.00

1N

M20.

131

.48

-0.2

731

.48

31.6

595

.16

1.00

222

.30

-0.0

222

.31

22.3

895

.10

1.00

2N

M20.

231

.48

-0.2

731

.48

31.6

695

.12

1.00

222

.31

-0.0

222

.31

22.3

995

.04

1.00

2N

M30.

131

.47

-0.2

331

.47

31.7

095

.04

1.00

322

.37

-0.0

122

.37

22.4

295

.28

1.00

3N

M30.

231

.50

-0.2

431

.50

31.7

294

.92

1.00

422

.39

-0.0

322

.40

22.4

395

.24

1.00

4N

M40.

131

.55

-0.2

531

.55

31.7

495

.16

1.00

422

.40

0.03

22.4

122

.44

95.0

61.

004

NM

40.

231

.59

-0.2

331

.59

31.7

995

.00

1.00

622

.43

0.02

22.4

422

.48

95.1

41.

006

NM

50.

131

.58

-0.1

831

.58

31.7

794

.98

1.00

522

.44

0.01

22.4

522

.46

95.0

21.

005

NM

50.

231

.71

-0.1

731

.71

31.8

994

.92

1.00

922

.54

-0.0

122

.55

22.5

495

.00

1.00

9N

M60.

131

.56

-0.2

531

.57

31.7

995

.06

1.00

622

.43

0.00

22.4

322

.48

95.0

21.

006

NM

60.

231

.78

-0.2

531

.78

31.9

995

.06

1.01

222

.54

-0.0

222

.54

22.6

295

.26

1.01

2N

M70.

131

.55

-0.1

731

.55

31.8

195

.24

1.00

722

.44

0.01

22.4

422

.49

95.2

01.

006

NM

70.

231

.87

-0.1

431

.88

32.1

495

.14

1.01

722

.65

-0.0

622

.65

22.7

295

.22

1.01

7N

M80.

131

.55

-0.1

931

.56

31.8

295

.14

1.00

722

.45

0.01

22.4

522

.50

94.9

01.

007

NM

80.

232

.04

-0.0

632

.04

32.3

194

.90

1.02

322

.75

-0.0

822

.75

22.8

495

.42

1.02

2N

M90.

131

.61

-0.1

631

.61

31.8

395

.06

1.00

722

.41

0.03

22.4

122

.50

95.1

01.

007

NM

90.

232

.24

-0.0

332

.25

32.5

295

.00

1.02

922

.83

-0.0

522

.83

22.9

995

.20

1.02

9

136

Tab

leE

77:

Infe

ren

cefo

rth

eva

rian

ceσ

2b

ased

onN

(µ=

0,σ

2=

1)d

ata

wit

hC

=1.

282

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

142.

17-1

1.97

141.

6813

9.73

92.6

81.

000

99.9

8-4

.90

99.8

799

.51

94.2

01.

000

MI.

C90

155.

16-1

0.84

154.

8014

0.92

91.1

21.

009

109.

31-4

.12

109.

2410

0.12

92.2

61.

006

MI.

C80

156.

750.

7315

6.76

143.

5191

.44

1.02

710

9.86

1.25

109.

8610

1.16

92.5

21.

017

MI.

D90

145.

12-2

.03

145.

1214

2.64

93.3

01.

021

100.

80-0

.66

100.

8110

0.53

94.1

01.

010

MI.

D80

144.

17-1

.47

144.

1814

2.86

93.5

21.

022

100.

790.

3910

0.80

100.

8894

.44

1.01

4C

EN

S154.

12-7

.99

153.

9315

2.24

92.9

81.

090

107.

73-3

.31

107.

6910

8.00

94.0

61.

085

NM

10.1

142.

67-1

1.54

142.

2114

0.36

92.7

61.

005

100.

56-4

.70

100.

4699

.94

93.8

41.

004

NM

10.2

142.

64-1

1.56

142.

1814

0.37

92.7

61.

005

100.

55-4

.70

100.

4599

.95

93.8

41.

004

NM

20.1

143.

73-1

1.16

143.

3114

1.82

92.9

41.

015

101.

28-4

.69

101.

1810

0.94

93.9

41.

014

NM

20.2

143.

80-1

1.15

143.

3814

1.90

92.8

81.

016

101.

33-4

.73

101.

2310

0.99

94.1

01.

015

NM

30.1

144.

80-1

0.88

144.

4014

3.49

93.1

21.

027

102.

66-4

.53

102.

5710

2.12

93.9

21.

026

NM

30.2

145.

16-1

1.05

144.

7514

3.77

93.1

01.

029

102.

92-4

.33

102.

8410

2.35

93.9

61.

029

NM

40.1

146.

57-9

.81

146.

2614

5.09

93.0

41.

038

102.

94-4

.49

102.

8510

3.11

93.8

41.

036

NM

40.2

147.

21-9

.92

146.

8914

5.87

93.0

61.

044

103.

38-4

.51

103.

2910

3.68

94.0

41.

042

NM

50.1

148.

10-1

0.19

147.

7614

6.08

92.8

01.

045

104.

19-5

.12

104.

0810

3.77

93.7

81.

043

NM

50.2

149.

44-1

0.30

149.

1014

7.76

93.2

21.

057

105.

36-5

.41

105.

2310

4.96

93.8

01.

055

NM

60.1

149.

12-1

0.06

148.

7914

6.82

93.1

61.

051

104.

17-3

.57

104.

1210

4.47

93.9

01.

050

NM

60.2

151.

85-1

0.64

151.

4914

9.78

93.0

81.

072

106.

21-3

.16

106.

1710

6.70

94.2

01.

072

NM

70.1

148.

79-9

.06

148.

5314

7.51

93.0

41.

056

104.

54-4

.13

104.

4710

4.76

93.9

81.

053

NM

70.2

152.

63-9

.54

152.

3415

2.19

92.9

41.

089

108.

24-3

.92

108.

1810

8.20

94.1

81.

087

NM

80.1

150.

20-9

.27

149.

9214

7.89

93.0

01.

058

104.

24-4

.05

104.

1810

5.06

94.3

21.

056

NM

80.2

156.

29-9

.55

156.

0215

4.48

93.4

81.

106

108.

49-4

.27

108.

4110

9.77

93.9

21.

103

NM

90.1

149.

59-9

.61

149.

3014

8.15

92.9

01.

060

105.

40-3

.61

105.

3510

5.33

94.0

61.

058

NM

90.2

156.

95-1

0.29

156.

6315

6.58

92.6

61.

121

111.

71-3

.66

111.

6611

1.42

94.0

41.

120

137

Tab

leE

78:

Infe

ren

cefo

rth

eva

rian

ceσ

2b

ased

onN

(µ=

0,σ

2=

1)d

ata

wit

hC

=1.

282

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

44.1

6-0

.92

44.1

544

.68

95.1

21.

000

31.2

1-1

.17

31.1

931

.59

95.1

81.

000

MI.

C90

48.2

1-0

.41

48.2

144

.87

93.4

01.

004

34.1

8-1

.24

34.1

631

.70

92.8

81.

004

MI.

C80

48.8

90.

6948

.89

45.0

992

.94

1.00

934

.48

-0.6

034

.48

31.8

392

.84

1.00

8M

I.D

90

44.4

2-0

.06

44.4

344

.90

95.2

81.

005

31.3

1-0

.84

31.3

031

.72

95.3

21.

004

MI.

D80

44.4

40.

0444

.45

44.9

995

.48

1.00

731

.37

-0.6

731

.37

31.7

895

.44

1.00

6C

EN

S47

.80

-0.3

247

.80

48.4

195

.52

1.08

334

.01

-0.9

434

.00

34.2

095

.12

1.08

3N

M10.

144

.37

-0.9

944

.36

44.8

695

.24

1.00

431

.36

-1.2

031

.34

31.7

295

.18

1.00

4N

M10.

244

.37

-1.0

044

.36

44.8

695

.30

1.00

431

.36

-1.2

031

.34

31.7

295

.18

1.00

4N

M20.

144

.77

-0.9

244

.76

45.3

195

.36

1.01

431

.63

-1.1

731

.61

32.0

395

.22

1.01

4N

M20.

244

.80

-0.9

244

.80

45.3

495

.36

1.01

531

.65

-1.1

831

.63

32.0

595

.22

1.01

5N

M30.

145

.34

-0.7

545

.34

45.8

495

.10

1.02

632

.13

-1.1

432

.12

32.4

095

.18

1.02

6N

M30.

245

.45

-0.7

845

.45

45.9

495

.08

1.02

832

.20

-1.1

832

.18

32.4

795

.18

1.02

8N

M40.

145

.56

-0.8

245

.56

46.2

995

.32

1.03

632

.36

-0.9

232

.35

32.7

395

.04

1.03

6N

M40.

245

.74

-0.7

845

.73

46.5

495

.36

1.04

232

.55

-0.9

632

.54

32.9

194

.86

1.04

2N

M50.

146

.31

-0.5

046

.32

46.6

395

.04

1.04

432

.65

-1.0

332

.64

32.9

595

.20

1.04

3N

M50.

246

.89

-0.4

946

.90

47.1

795

.08

1.05

633

.01

-1.1

032

.99

33.3

495

.06

1.05

5N

M60.

146

.18

-0.8

246

.18

46.8

495

.32

1.04

832

.85

-1.0

632

.83

33.1

195

.04

1.04

8N

M60.

247

.00

-0.8

447

.00

47.8

295

.20

1.07

033

.65

-1.1

333

.64

33.8

194

.88

1.07

0N

M70.

146

.38

-0.5

246

.39

47.0

295

.46

1.05

233

.15

-1.0

233

.14

33.2

395

.10

1.05

2N

M70.

248

.11

-0.4

248

.11

48.5

695

.26

1.08

734

.16

-1.1

934

.14

34.3

194

.88

1.08

6N

M80.

146

.53

-0.5

646

.53

47.1

495

.54

1.05

533

.14

-1.0

233

.12

33.3

195

.18

1.05

5N

M80.

248

.34

-0.2

648

.34

49.2

995

.48

1.10

334

.62

-1.2

934

.60

34.8

294

.88

1.10

2N

M90.

146

.66

-0.4

946

.67

47.2

495

.34

1.05

733

.07

-0.9

333

.06

33.3

995

.18

1.05

7N

M90.

249

.63

-0.1

749

.64

50.0

195

.12

1.11

935

.02

-1.1

035

.01

35.3

394

.72

1.11

9

138

Tab

leE

79:

Infe

ren

cefo

rth

e84

thp

erce

nti

leµ

+σ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

282

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

121.8

1-9

.72

121.

4312

1.43

94.5

41.

000

85.5

5-5

.82

85.3

686

.28

95.0

81.

000

MI.

C90

131.2

5-1

1.21

130.

7812

1.81

92.9

61.

003

92.8

9-6

.36

92.6

886

.55

92.7

61.

003

MI.

C80

134.7

0-0

.67

134.

7112

3.03

92.5

21.

013

94.2

0-1

.71

94.1

987

.14

92.7

61.

010

MI.

D90

122.3

1-5

.93

122.

1812

2.54

94.7

61.

009

85.8

0-4

.21

85.7

186

.79

95.1

81.

006

MI.

D80

122.4

9-6

.03

122.

3512

2.95

94.8

61.

013

85.7

8-3

.87

85.7

087

.11

95.2

21.

010

CE

NS

128.8

3-7

.23

128.

6412

9.22

94.7

61.

064

90.4

7-4

.87

90.3

591

.64

94.9

41.

062

NM

10.1

122.2

9-9

.44

121.

9312

1.81

94.4

81.

003

85.8

9-5

.70

85.7

186

.55

94.7

41.

003

NM

10.2

122.2

7-9

.46

121.

9212

1.82

94.5

01.

003

85.9

2-5

.69

85.7

486

.55

94.7

21.

003

NM

20.1

122.7

2-9

.20

122.

3912

2.73

94.6

21.

011

86.4

3-5

.73

86.2

587

.18

94.8

81.

010

NM

20.2

122.8

1-9

.18

122.

4812

2.80

94.7

61.

011

86.5

2-5

.77

86.3

487

.23

94.8

61.

011

NM

30.1

123.3

8-9

.05

123.

0612

3.79

94.7

41.

019

87.0

7-5

.62

86.9

087

.93

94.9

41.

019

NM

30.2

123.5

3-9

.22

123.

2012

4.05

94.8

41.

022

87.3

8-5

.45

87.2

288

.13

94.9

41.

021

NM

40.1

124.4

8-8

.30

124.

2212

4.77

94.7

61.

028

87.4

4-5

.62

87.2

788

.57

95.0

41.

026

NM

40.2

125.1

9-8

.43

124.

9212

5.48

94.7

41.

033

88.0

4-5

.65

87.8

689

.08

95.1

01.

032

NM

50.1

125.8

7-8

.65

125.

5912

5.41

94.6

21.

033

88.0

5-6

.12

87.8

489

.00

94.9

01.

032

NM

50.2

127.3

6-8

.79

127.

0712

6.92

94.3

01.

045

89.1

4-6

.40

88.9

290

.07

94.9

61.

044

NM

60.1

125.8

4-8

.59

125.

5612

5.87

94.5

01.

037

88.3

6-4

.95

88.2

389

.41

94.9

61.

036

NM

60.2

128.5

7-9

.20

128.

2512

8.57

94.7

61.

059

90.4

4-4

.65

90.3

391

.38

95.0

81.

059

NM

70.1

126.4

0-7

.81

126.

1712

6.29

94.8

41.

040

88.2

7-5

.40

88.1

189

.61

94.8

01.

039

NM

70.2

129.5

1-8

.37

129.

2513

0.56

94.9

41.

075

91.5

6-5

.38

91.4

192

.70

94.5

81.

074

NM

80.1

126.2

9-8

.02

126.

0512

6.53

94.7

41.

042

88.4

6-5

.32

88.3

189

.80

94.8

41.

041

NM

80.2

131.9

3-8

.49

131.

6713

2.66

94.4

81.

093

92.6

8-5

.64

92.5

294

.17

94.9

01.

091

NM

90.1

125.9

3-8

.26

125.

6712

6.71

94.7

61.

044

89.0

1-5

.04

88.8

889

.96

94.8

21.

043

NM

90.2

132.4

6-9

.19

132.

1613

4.83

94.7

41.

110

95.5

2-5

.26

95.3

995

.77

94.7

21.

110

139

Tab

leE

80:

Infe

ren

cefo

rth

e84

thp

erce

nti

leµ

+σ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

282

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

38.3

9-0

.98

38.3

838

.70

95.3

21.

000

27.2

8-0

.73

27.2

727

.37

95.2

01.

000

MI.

C90

41.2

1-0

.80

41.2

138

.82

93.3

21.

003

29.2

7-0

.92

29.2

527

.44

93.5

61.

003

MI.

C80

42.5

50.

1342

.56

39.0

092

.80

1.00

830

.06

-0.3

330

.06

27.5

692

.90

1.00

7M

I.D

90

38.5

6-0

.61

38.5

638

.85

95.3

61.

004

27.3

6-0

.62

27.3

627

.47

95.1

61.

004

MI.

D80

38.7

1-0

.67

38.7

138

.98

95.2

01.

007

27.4

2-0

.54

27.4

227

.56

95.2

01.

007

CE

NS

40.6

5-0

.58

40.6

541

.08

95.1

81.

061

28.9

8-0

.58

28.9

829

.04

95.1

41.

061

NM

10.

138

.59

-1.0

338

.58

38.8

295

.34

1.00

327

.37

-0.7

527

.36

27.4

595

.22

1.00

3N

M10.

238

.59

-1.0

438

.58

38.8

295

.30

1.00

327

.37

-0.7

527

.36

27.4

595

.18

1.00

3N

M20.

138

.85

-0.9

838

.84

39.1

095

.42

1.01

027

.49

-0.7

327

.49

27.6

595

.26

1.01

0N

M20.

238

.87

-0.9

838

.86

39.1

395

.40

1.01

127

.51

-0.7

427

.51

27.6

795

.24

1.01

1N

M30.

139

.08

-0.8

739

.07

39.4

495

.30

1.01

927

.86

-0.7

127

.85

27.8

995

.18

1.01

9N

M30.

239

.18

-0.8

939

.17

39.5

395

.30

1.02

127

.94

-0.7

527

.94

27.9

595

.14

1.02

1N

M40.

139

.36

-0.9

239

.36

39.7

395

.14

1.02

628

.04

-0.5

628

.04

28.0

994

.78

1.02

7N

M40.

239

.52

-0.8

839

.51

39.9

695

.18

1.03

228

.20

-0.5

928

.19

28.2

594

.92

1.03

2N

M50.

139

.74

-0.6

939

.74

39.9

494

.96

1.03

228

.28

-0.6

428

.28

28.2

494

.60

1.03

2N

M50.

240

.28

-0.6

940

.28

40.4

394

.96

1.04

528

.64

-0.7

028

.64

28.5

894

.72

1.04

4N

M60.

139

.67

-0.9

239

.66

40.0

895

.12

1.03

628

.31

-0.6

628

.31

28.3

495

.00

1.03

6N

M60.

240

.50

-0.9

540

.49

40.9

695

.02

1.05

828

.92

-0.7

328

.91

28.9

695

.04

1.05

8N

M70.

139

.69

-0.7

039

.69

40.1

995

.42

1.03

828

.48

-0.6

428

.47

28.4

295

.02

1.03

8N

M70.

241

.16

-0.6

441

.16

41.5

895

.28

1.07

429

.37

-0.8

029

.36

29.3

994

.80

1.07

4N

M80.

139

.78

-0.7

439

.78

40.2

795

.26

1.04

128

.49

-0.6

328

.48

28.4

795

.06

1.04

0N

M80.

241

.57

-0.4

841

.58

42.2

695

.00

1.09

229

.78

-0.8

829

.77

29.8

794

.90

1.09

1N

M90.

139

.97

-0.6

839

.97

40.3

495

.32

1.04

228

.37

-0.5

728

.37

28.5

295

.06

1.04

2N

M90.

242

.61

-0.4

342

.61

42.9

795

.30

1.11

030

.11

-0.7

530

.10

30.3

895

.18

1.11

0

140

Tab

leE

81:

Infe

ren

cefo

rth

e95

thp

erce

nti

leµ

+1.6

45σ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

282

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103

×10

3%

Len

.×

103×

103

×10

3×

103

%L

en.

CD

153.

85-1

5.24

153.

1115

2.08

93.9

01.

000

107.

48-8

.21

107.

1810

8.06

94.7

01.

000

MI.

C90

168.

45-1

7.25

167.

5815

2.68

92.0

01.

004

118.

68-8

.94

118.

3510

8.48

92.3

01.

004

MI.

C80

171.

95-3

.51

171.

9315

4.29

91.7

01.

015

120.

05-2

.82

120.

0310

9.28

92.2

01.

011

MI.

D90

154.

48-9

.02

154.

2415

3.62

94.1

01.

010

107.

90-5

.56

107.

7710

8.78

94.7

01.

007

MI.

D80

154.

80-9

.10

154.

5515

4.12

94.3

01.

013

107.

85-5

.01

107.

7510

9.20

95.0

01.

010

CE

NS

165.

22-1

1.73

164.

8216

4.68

94.2

01.

083

115.

48-6

.88

115.

2811

6.77

94.6

01.

081

NM

10.1

154.

53-1

4.84

153.

8315

2.71

93.9

01.

004

108.

05-8

.04

107.

7610

8.50

94.4

01.

004

NM

10.2

154.

51-1

4.85

153.

8115

2.72

93.9

01.

004

108.

08-8

.03

107.

7910

8.51

94.4

01.

004

NM

20.1

155.

33-1

4.49

154.

6715

4.19

93.9

01.

014

108.

90-8

.07

108.

6110

9.54

94.7

01.

014

NM

20.2

155.

45-1

4.47

154.

7915

4.30

93.9

01.

015

109.

02-8

.13

108.

7210

9.61

94.6

01.

014

NM

30.1

156.

44-1

4.28

155.

8115

5.92

94.3

01.

025

110.

05-7

.94

109.

7711

0.76

94.5

01.

025

NM

30.2

156.

76-1

4.52

156.

1015

6.31

94.2

01.

028

110.

47-7

.70

110.

2111

1.05

94.6

01.

028

NM

40.1

158.

15-1

3.22

157.

6115

7.50

94.2

01.

036

110.

56-7

.93

110.

2911

1.80

94.9

01.

035

NM

40.2

159.

16-1

3.40

158.

6115

8.55

94.4

01.

043

111.

37-7

.97

111.

1011

2.55

94.9

01.

042

NM

50.1

160.

22-1

3.73

159.

6515

8.55

94.0

01.

042

111.

68-8

.65

111.

3611

2.51

94.5

01.

041

NM

50.2

162.

33-1

3.94

161.

7416

0.79

93.8

01.

057

113.

30-9

.05

112.

9511

4.10

94.8

01.

056

NM

60.1

160.

45-1

3.64

159.

8815

9.30

94.3

01.

047

111.

97-6

.98

111.

7711

3.15

94.5

01.

047

NM

60.2

164.

43-1

4.52

163.

8016

3.28

94.1

01.

074

114.

94-6

.58

114.

7611

6.07

94.9

01.

074

NM

70.1

160.

96-1

2.53

160.

4915

9.94

94.2

01.

052

112.

02-7

.62

111.

7711

3.49

94.7

01.

050

NM

70.2

165.

74-1

3.35

165.

2216

6.23

94.3

01.

093

116.

89-7

.59

116.

6611

8.03

94.4

01.

092

NM

80.1

161.

22-1

2.84

160.

7316

0.35

94.3

01.

054

112.

13-7

.51

111.

8911

3.79

94.8

01.

053

NM

80.2

169.

49-1

3.55

168.

9616

9.28

94.5

01.

113

118.

21-7

.97

117.

9512

0.16

94.8

01.

112

NM

90.1

160.

68-1

3.18

160.

1516

0.64

94.2

01.

056

113.

10-7

.11

112.

8911

4.05

94.7

01.

055

NM

90.2

170.

41-1

4.52

169.

8117

2.36

94.4

01.

133

122.

31-7

.45

122.

1012

2.44

94.5

01.

133

141

Tab

leE

82:

Infe

ren

cefo

rth

e95

thp

erce

nti

leµ

+1.6

45σ

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

282

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

48.0

3-1

.43

48.0

148

.47

95.5

01.

000

34.1

0-1

.19

34.0

834

.28

95.3

01.

000

MI.

C90

52.5

0-1

.18

52.4

948

.66

92.8

01.

004

37.2

9-1

.44

37.2

634

.40

93.0

01.

004

MI.

C80

54.1

30.

0554

.13

48.9

192

.60

1.00

938

.21

-0.6

738

.21

34.5

792

.60

1.00

8M

I.D

90

48.3

0-0

.85

48.2

948

.70

95.5

01.

005

34.2

2-1

.00

34.2

134

.43

95.5

01.

004

MI.

D80

48.4

6-0

.90

48.4

648

.86

95.4

01.

008

34.3

0-0

.88

34.3

034

.54

95.4

01.

008

CE

NS

51.7

4-0

.86

51.7

452

.34

95.0

01.

080

36.9

1-0

.98

36.9

037

.00

95.1

01.

079

NM

10.

148

.32

-1.5

148

.30

48.6

695

.40

1.00

434

.25

-1.2

234

.23

34.4

195

.20

1.00

4N

M10.

248

.32

-1.5

248

.30

48.6

795

.30

1.00

434

.25

-1.2

134

.23

34.4

195

.20

1.00

4N

M20.

148

.75

-1.4

448

.73

49.1

395

.40

1.01

434

.47

-1.1

834

.46

34.7

495

.40

1.01

4N

M20.

248

.78

-1.4

448

.77

49.1

795

.40

1.01

434

.50

-1.2

034

.49

34.7

795

.40

1.01

4N

M30.

149

.19

-1.2

849

.18

49.6

895

.30

1.02

535

.05

-1.1

635

.03

35.1

395

.50

1.02

5N

M30.

249

.34

-1.3

149

.32

49.8

195

.20

1.02

835

.17

-1.2

135

.15

35.2

295

.40

1.02

7N

M40.

149

.59

-1.3

649

.57

50.1

595

.00

1.03

535

.33

-0.9

435

.32

35.4

694

.90

1.03

5N

M40.

249

.81

-1.3

049

.80

50.4

995

.10

1.04

235

.56

-0.9

835

.55

35.7

094

.90

1.04

2N

M50.

150

.24

-1.0

350

.24

50.5

095

.20

1.04

235

.69

-1.0

635

.68

35.7

094

.90

1.04

2N

M50.

251

.03

-1.0

351

.03

51.2

295

.20

1.05

736

.21

-1.1

436

.20

36.2

195

.00

1.05

6N

M60.

150

.13

-1.3

650

.12

50.7

295

.20

1.04

635

.79

-1.0

935

.78

35.8

795

.00

1.04

6N

M60.

251

.32

-1.4

051

.30

52.0

295

.10

1.07

336

.72

-1.1

936

.71

36.7

894

.90

1.07

3N

M70.

150

.21

-1.0

450

.20

50.9

095

.30

1.05

036

.07

-1.0

636

.06

35.9

995

.20

1.05

0N

M70.

252

.40

-0.9

652

.40

52.9

595

.20

1.09

237

.39

-1.2

837

.37

37.4

395

.10

1.09

2N

M80.

150

.35

-1.0

950

.35

51.0

395

.20

1.05

336

.08

-1.0

536

.06

36.0

895

.00

1.05

3N

M80.

252

.93

-0.7

552

.93

53.9

395

.00

1.11

337

.99

-1.3

937

.97

38.1

195

.00

1.11

2N

M90.

150

.61

-1.0

250

.61

51.1

495

.30

1.05

535

.93

-0.9

535

.92

36.1

595

.20

1.05

5N

M90.

254

.49

-0.6

854

.49

54.9

595

.20

1.13

438

.48

-1.2

138

.47

38.8

495

.30

1.13

3

142

Tab

leE

83:

Infe

ren

cefo

rµ

+σ

2/2

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

282

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

0R

esu

lts

forn

=20

0

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103×

103

×10

3×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

121

.43

-7.1

512

1.23

121.

3694

.50

1.00

085

.40

-4.5

685

.28

86.2

795

.00

1.00

0M

I.C

90

131

.10

-7.2

813

0.91

121.

9593

.10

1.00

592

.81

-4.4

392

.71

86.6

192

.80

1.00

4M

I.C

80

135

.17

4.09

135.

1312

3.55

92.6

01.

018

94.2

90.

6594

.30

87.3

392

.80

1.01

2M

I.D

90

122

.75

-2.1

612

2.75

122.

9794

.90

1.01

385

.79

-2.4

685

.76

86.9

095

.20

1.00

7M

I.D

80

122

.62

-2.0

112

2.61

123.

3494

.90

1.01

685

.79

-1.9

185

.78

87.2

695

.30

1.01

1C

EN

S128

.91

-4.2

512

8.85

129.

3794

.70

1.06

690

.39

-3.4

290

.34

91.7

095

.00

1.06

3N

M10.

1121

.92

-6.8

612

1.74

121.

7594

.50

1.00

385

.75

-4.4

385

.64

86.5

494

.70

1.00

3N

M10.

2121

.90

-6.8

712

1.72

121.

7694

.60

1.00

385

.77

-4.4

285

.67

86.5

594

.70

1.00

3N

M20.

1122

.41

-6.5

712

2.25

122.

6894

.70

1.01

186

.29

-4.4

486

.19

87.1

894

.90

1.01

0N

M20.

2122

.50

-6.5

612

2.33

122.

7594

.90

1.01

186

.38

-4.4

786

.28

87.2

295

.00

1.01

1N

M30.

1123

.05

-6.3

912

2.90

123.

7594

.80

1.02

086

.95

-4.3

086

.85

87.9

394

.90

1.01

9N

M30.

2123

.19

-6.5

512

3.03

124.

0194

.90

1.02

287

.26

-4.1

187

.18

88.1

395

.00

1.02

2N

M40.

1124

.27

-5.5

912

4.16

124.

7794

.70

1.02

887

.31

-4.2

987

.21

88.5

795

.10

1.02

7N

M40.

2124

.95

-5.6

912

4.84

125.

4794

.70

1.03

487

.91

-4.3

187

.81

89.0

895

.10

1.03

3N

M50.

1125

.62

-5.8

812

5.50

125.

4294

.70

1.03

387

.89

-4.7

587

.77

89.0

094

.90

1.03

2N

M50.

2127

.07

-5.9

612

6.95

126.

9294

.50

1.04

688

.98

-5.0

088

.85

90.0

794

.90

1.04

4N

M60.

1125

.68

-5.7

812

5.56

125.

9094

.60

1.03

788

.26

-3.5

988

.19

89.4

494

.90

1.03

7N

M60.

2128

.35

-6.2

912

8.21

128.

5694

.80

1.05

990

.35

-3.2

390

.30

91.4

295

.20

1.06

0N

M70.

1126

.25

-5.0

212

6.16

126.

3494

.90

1.04

188

.14

-4.0

388

.05

89.6

494

.80

1.03

9N

M70.

2129

.29

-5.4

212

9.19

130.

5994

.90

1.07

691

.43

-3.9

091

.36

92.7

394

.70

1.07

5N

M80.

1126

.23

-5.1

812

6.14

126.

5994

.80

1.04

388

.34

-3.9

688

.26

89.8

394

.90

1.04

1N

M80.

2131

.81

-5.4

113

1.71

132.

7194

.50

1.09

492

.57

-4.1

692

.49

94.1

994

.90

1.09

2N

M90.

1125

.77

-5.4

412

5.67

126.

7694

.60

1.04

588

.93

-3.6

588

.86

90.0

094

.80

1.04

3N

M90.

2132

.20

-6.0

713

2.07

134.

8694

.80

1.11

195

.46

-3.6

995

.39

95.8

294

.70

1.11

1

143

Tab

leE

84:

Infe

ren

cefo

rµ

+σ

2/2

bas

edon

N(µ

=0,σ

2=

1)d

ata

wit

hC

=1.

282

=90

thp

erce

nti

le

Res

ult

sfo

rn

=10

00R

esu

lts

forn

=20

00

RM

SE

Bia

sS

DS

DC

vg.

Rel

.R

MS

EB

ias

SD

SD

Cvg.

Rel

.×

103

×10

3×

103×

103

%L

en.

×10

3×

103×

103×

103

%L

en.

CD

38.3

8-0

.73

38.3

738

.70

95.3

01.

000

27.2

7-0

.61

27.2

627

.36

95.2

01.

000

MI.

C90

41.2

1-0

.43

41.2

138

.83

93.4

01.

003

29.2

5-0

.73

29.2

527

.44

93.6

01.

003

MI.

C80

42.5

70.

6042

.57

39.0

292

.80

1.00

830

.05

-0.1

030

.05

27.5

692

.90

1.00

7M

I.D

90

38.5

6-0

.27

38.5

638

.86

95.4

01.

004

27.3

5-0

.46

27.3

527

.47

95.1

01.

004

MI.

D80

38.7

1-0

.29

38.7

138

.99

95.2

01.

008

27.4

2-0

.35

27.4

227

.56

95.2

01.

007

CE

NS

40.6

5-0

.29

40.6

541

.08

95.2

01.

062

28.9

7-0

.44

28.9

729

.04

95.1

01.

061

NM

10.

138

.57

-0.7

838

.57

38.8

295

.40

1.00

327

.35

-0.6

327

.35

27.4

595

.20

1.00

3N

M10.

238

.57

-0.7

938

.57

38.8

295

.30

1.00

327

.35

-0.6

327

.35

27.4

595

.20

1.00

3N

M20.

138

.83

-0.7

338

.83

39.1

095

.40

1.01

027

.48

-0.6

027

.48

27.6

595

.20

1.01

0N

M20.

238

.85

-0.7

338

.85

39.1

395

.50

1.01

127

.50

-0.6

127

.50

27.6

695

.30

1.01

1N

M30.

139

.07

-0.6

139

.07

39.4

495

.30

1.01

927

.85

-0.5

827

.84

27.8

895

.20

1.01

9N

M30.

239

.17

-0.6

339

.17

39.5

395

.30

1.02

127

.93

-0.6

227

.93

27.9

495

.10

1.02

1N

M40.

139

.35

-0.6

639

.35

39.7

395

.20

1.02

728

.04

-0.4

328

.04

28.0

994

.80

1.02

7N

M40.

239

.50

-0.6

239

.50

39.9

695

.20

1.03

228

.19

-0.4

628

.19

28.2

594

.90

1.03

2N

M50.

139

.74

-0.4

339

.74

39.9

594

.90

1.03

228

.27

-0.5

028

.27

28.2

494

.60

1.03

2N

M50.

240

.28

-0.4

240

.28

40.4

395

.00

1.04

528

.63

-0.5

628

.62

28.5

894

.70

1.04

4N

M60.

139

.66

-0.6

639

.66

40.0

895

.10

1.03

628

.30

-0.5

328

.30

28.3

495

.00

1.03

6N

M60.

240

.49

-0.6

740

.49

40.9

695

.00

1.05

828

.90

-0.5

928

.90

28.9

695

.10

1.05

8N

M70.

139

.69

-0.4

339

.70

40.2

095

.40

1.03

928

.47

-0.5

028

.47

28.4

195

.10

1.03

8N

M70.

241

.16

-0.3

541

.17

41.5

995

.30

1.07

529

.36

-0.6

629

.36

29.3

994

.80

1.07

4N

M80.

139

.77

-0.4

639

.77

40.2

895

.20

1.04

128

.48

-0.4

928

.48

28.4

795

.10

1.04

0N

M80.

241

.57

-0.1

941

.57

42.2

695

.00

1.09

229

.77

-0.7

329

.76

29.8

694

.90

1.09

1N

M90.

139

.96

-0.4

139

.97

40.3

495

.40

1.04

228

.36

-0.4

328

.36

28.5

295

.10

1.04

2N

M90.

242

.61

-0.1

242

.62

42.9

895

.20

1.11

130

.09

-0.6

030

.09

30.3

795

.20

1.11

0

144

APPENDIX F: ANALYSIS OF FULLY MASKED DATA ON TOTAL HOUSEHOLD INCOMEAND HOUSEHOLD ALIMONEY PAYEMENT BASED ON DATA FROM THE 2000 U.S.

CURRENT POPULATION SURVEY

In this appendix we present tables summarizing analysis of two variables of the 2000 U.S.Current Population Survey data under full data masking. The data and notations appearing in thetables are described in Section 7 of the paper.

Table F1. Analysis of household total income under full masking



CD 10.495 1.000 0.981 1.000 58996 1.000 5805.083 1.000 97263 1.000 184262 1.000MI 10.495 1.015 0.980 1.013 58977 1.016 5793.485 1.014 97231 1.016 184145 1.015NM10U 10.494 1.001 0.980 1.003 58958 1.001 5790.372 1.000 97201 1.001 184097 1.001NM10C 10.494 1.002 0.981 1.003 58962 1.001 5795.846 1.001 97207 1.002 184139 1.002NM20U 10.495 1.008 0.983 1.015 59041 1.011 5827.769 1.017 97338 1.011 184492 1.013NM20C 10.496 1.006 0.980 1.012 59048 1.009 5804.365 1.011 97349 1.009 184354 1.011NM30U 10.493 1.015 0.980 1.028 58879 1.017 5769.845 1.019 97071 1.018 183817 1.021NM30C 10.493 1.016 0.984 1.033 59003 1.022 5833.608 1.033 97277 1.022 184462 1.026NM40U 10.493 1.032 0.987 1.059 59074 1.042 5872.531 1.063 97394 1.041 184845 1.049NM40C 10.493 1.027 0.982 1.054 58932 1.035 5803.355 1.047 97159 1.035 184134 1.041NM50U 10.495 1.048 0.983 1.087 59057 1.061 5829.819 1.082 97364 1.061 184535 1.071NM50C 10.494 1.038 0.978 1.078 58889 1.050 5755.189 1.061 97086 1.051 183738 1.060NM60U 10.489 1.076 0.990 1.139 58936 1.093 5878.810 1.134 97168 1.092 184631 1.109NM60C 10.493 1.056 0.980 1.115 58892 1.074 5775.893 1.096 97092 1.074 183881 1.088NM70U 10.485 1.108 0.991 1.201 58741 1.127 5848.492 1.179 96847 1.126 184076 1.150NM70C 10.497 1.075 0.983 1.156 59196 1.106 5861.327 1.148 97595 1.106 184997 1.126NM80U 10.483 1.149 0.984 1.277 58404 1.160 5710.635 1.210 96288 1.160 182554 1.191NM80C 10.492 1.095 0.984 1.200 58919 1.129 5812.647 1.177 97138 1.129 184171 1.155NM90U 10.481 1.198 0.951 1.367 57307 1.165 5216.701 1.163 94454 1.173 177174 1.205NM90C 10.490 1.113 0.977 1.239 58616 1.149 5691.716 1.189 96635 1.150 182819 1.180

145

Table F2. Analysis of household alimony payments under full masking



CD 8.715 1.000 1.168 1.000 10930 1.000 264.812 1.000 17962 1.000 36069 1.000MI 8.724 1.004 1.147 1.012 10988 1.037 280.919 1.329 18013 1.027 36058 1.039NM10U 8.709 1.000 1.165 1.000 10847 0.993 259.675 0.981 17828 0.993 35767 0.993NM10C 8.715 1.003 1.172 1.006 10947 1.006 267.222 1.014 17987 1.005 36164 1.007NM20U 8.724 1.013 1.185 1.026 11115 1.035 280.355 1.079 18255 1.031 36834 1.037NM20C 8.707 1.006 1.169 1.012 10843 1.000 260.912 0.996 17818 1.000 35788 1.001NM30U 8.720 1.035 1.220 1.071 11272 1.081 303.136 1.204 18484 1.069 37683 1.089NM30C 8.702 1.005 1.151 1.011 10695 0.986 247.352 0.946 17587 0.989 35136 0.987NM40U 8.709 1.011 1.135 1.021 10690 0.991 241.386 0.934 17587 0.997 34968 0.993NM40C 8.712 1.028 1.182 1.057 10977 1.043 272.638 1.078 18030 1.039 36358 1.049NM50U 8.705 1.027 1.137 1.053 10650 1.008 240.305 0.955 17520 1.013 34857 1.013NM50C 8.738 1.025 1.147 1.050 11058 1.046 262.552 1.040 18185 1.049 36280 1.051NM60U 8.737 1.099 1.266 1.207 11733 1.219 350.650 1.533 19194 1.192 39661 1.242NM60C 8.704 1.020 1.102 1.040 10458 0.983 219.756 0.872 17220 0.994 33890 0.986NM70U 8.688 1.100 1.197 1.203 10795 1.112 269.324 1.183 17719 1.102 35888 1.133NM70C 8.712 1.063 1.168 1.129 10896 1.084 262.985 1.107 17906 1.082 35951 1.098NM80U 8.667 1.168 1.273 1.345 10981 1.214 310.293 1.475 17956 1.183 37181 1.249NM80C 8.735 1.058 1.116 1.120 10863 1.075 242.053 1.024 17881 1.084 35339 1.088NM90U 8.694 1.230 1.294 1.491 11397 1.331 343.876 1.762 18614 1.287 38769 1.383NM90C 8.738 1.141 1.282 1.302 11842 1.303 365.256 1.710 19354 1.267 40176 1.337

146

a comparison of statistical disclosure control methods: multiple

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