combination of levene-type tests and a finite-intersection method for testing equality ... ·...

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UNIVERSITY OF WATERLOO

DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE

WORKING PAPER SERIES

2010-02

Combination of Levene-type Tests anda Finite-intersection Method for

Testing Equality of Variances againstOrdered Alternatives

Kimihiro Noguchi,University of California, Davis, e-mail:

[email protected]

Yulia R. Gel,University of Waterloo, e-mail:

[email protected]

Abstract

The problem of detecting monotonic trends in variances from k samplesis widely met in many applications, e.g. finance, economics, medicine, bio-pharmaceutical and environmental studies. However, most of the tests forequality of variances against ordered alternatives rely on the assumption ofnormality and are often non-robust to its violation, which eventually leadsto unreliable conclusions. In this paper, we propose a new distribution-freetest against trends in variances which is based on a combination of a robustLevene-type approach and a finite-intersection method. The new test can beviewed as a piece-wise linear approximation to possibly non-linear dynam-ics of variances, and hence can be applicable to a broad range of alternatives.The new combined procedure yields a more accurate estimate of a size andprovides a competitive power for a variety of distributions and alternatives.In addition, we develop a modification of the proposed test for unbalanceddesigns with unequal sample sizes. We discuss asymptotic properties of thenew test and illustrate its application by simulations and case studies from soilpollution analysis, real estate market, engineering and biology.

1 Introduction

The problem of equality of variances from k samples is widely met in many ap-plications (Conover et. al., 1981; Underwood, 1997) and there exist numerous sta-tistical procedures for assessing such problem (Lim and Loh, 1996; Zhang, 1998).Bartlett’s (1937) test, one of the first tests for equality of variances, assumes thatthe samples are normally distributed. The test is known to be sensitive to evenminor departures from normality (Box, 1953; Manoukian et al., 1986). Neverthe-less, Bartlett’s test is still widely employed in many studies even when normalityof samples is questionable (Mitchell, 1981). Levene (1960) proposes a new test forequality of variances using transformed data, Zij = |Xij − Xi|, the absolute devi-ation of Xij from the i-th sample mean, Xi. Levene (1960) applies an F -statisticto Zij to obtain the statistic. Due to its robustness to deviations from normalityand competitive power, Levene’s test has become very popular nowadays, andthe distribution-free Levene’s approach and its numerous robust modifications arewidely accepted in statistics and applied disciplines (Neuhauser, 2007).

However, the main question of interest often becomes not simply to assess het-erogeneity in variances, but to test whether variances follow a certain pattern asspecific configurations in data variability are suspected in many applications. Forexample, we frequently observe increasing or decreasing patterns in volatility ofstocks, or variability of air pollutant concentration as a function of proximity tothe sources of pollution. In general, the existing tests for detecting such a mono-tonic trend in variability can be divided into three main classes. The precursoryapproaches by Boswell and Brunk (1969) and Fujino (1979) employ the order-restricted maximum likelihood estimates of the variances (Brunk, 1955) that areapplied to the classical tests for equality of variances such as Bartlett’s or Cochran’s(1941) tests. The second class of tests is based on regression of variances on certain

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preselected scores (Vincent, 1961; Fujino, 1979). The third approach is called a‘finite-intersection approach’ due to Mudholkar et al. (1993, 1995). A test basedon the finite-intersection approach combines p-values for component test statisticswhich correspond to a finite number of nested hypotheses using various p-valuecombining methods. The idea of nested hypotheses is originally suggested byHogg (1961), and is found to be applicable to a variety of other hypothesis testingproblems (Hogg, 1962; Hogg and Tanis, 1963; Natarajan et al., 2005). Mudholkaret al. (1993, 1995) show that the finite-intersection tests provide competitive powercompared to the previously introduced tests.

However, many of the above tests for checking trends in variances assume thatthe observed data are normally distributed, and such tests exhibit substantial sen-sitivity to deviations from normality. To construct a robust test, Levene’s trans-formation Zij = |Xij − Xi| can be utilized. For example, Neuhauser and Hothorn(2000) propose an application of Bartholomew’s (1961) test and the double contrasttest using contrast vectors suggested by Bechhofer and Dunnett (1982) to Zij whileHui et al. (2008) employ a simple linear regression of Zij on linear scores. Suchprocedures based on Levene’s transformation generally yield a correct size of thetest and competitive power under normal, heavy-tailed, and skewed distributions.

In this paper, we propose a new distribution-free test for equality of variancesagainst ordered alternatives by combining the regression statistic of Hui et. al.(2008) and the finite-intersection method by Mudholkar et al. (1993, 1995). Thenew finite-intersection test can be viewed as a piecewise linear approximations ofan unknown (possibly nonlinear) trend in variances while Levene’s transforma-tion makes the test more robust under a general class of distributions. We showthat each component test statistic asymptotically follows the standard normal dis-tribution under the assumption of equality of variances. The joint distribution ofthe component test statistics are asymptotically multivariate standard normal un-der the same assumption, from which we deduce the asymptotic independence ofthe component test statistics. Our Monte Carlo simulations indicate that the newtest yields the correct size of the test for skewed and heavy-tailed distributionsand provides competitive power. Also, we present several modifications of thetest using O’Brien’s (1978) correction factor and the Hines-Hines (2000) structuralzero removal method for unbalanced designs or small sample sizes. Finally, we il-lustrate applications of the new test with real-life examples from microeconomics,engineering, environmental studies, and health studies.

The article is organized as follows. In Section 2, we review a brief historyand concept of nested hypotheses. Nested hypotheses are linked to the idea ofthe finite-intersection method, which allows different ways of combining p-valuesfrom the component test statistics. Also, we describe four p-value combining tech-niques for the finite-intersection method.

In Section 3, we introduce the new test, which combines the idea of nestedhypotheses and Levene’s transformation. We discuss the formulation of the com-ponent test statistics and the asymptotic properties, in particular, asymptotic nor-mality and independence. We use these asymptotic properties as a justification forusing the finite-intersection method.

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In Section 4, we discuss modifications of the new test for unequal or small sam-ple sizes. Specifically, we introduce a new correction factor for such cases basedon a combination of the Hines-Hines (2000) structural zero removal method andO’Brien’s (1978) correction factor.

In Section 5, we present a Monte Carlo simulation study to check the robustnessand power of the new test against previously proposed tests. We show that thenew test with the new correction factor is robust and frequently most powerfulbased on the simulation study.

In Section 6, we discuss applications of the tests for equality of variances againstordered alternatives. The real-life examples are taken from diverse disciplines,including real estate prices, engineering, environmental and health studies. Weshow that the new test is widely applicable for both normal and non-normal dis-tributions, and balanced and unbalanced designs.

We summarize the main results and an outline of the future work in Section 7.

2 Review of Nested Hypotheses

An intuitive approach to solve a complex problem is to “divide each of the difficul-ties under examination into as many parts as possible” as Descartes (1637) statesin Discourse on the Method. The concept of nested hypotheses, first suggested in aformal manner by Hogg (1961), follows the philosophy of Descartes by breakingdown the original hypothesis H0 into a finite number of more manageable compo-nents.

The concept of nested hypotheses is described as follows. Let Ω be a nonemptyparameter subspace of the parameter θ, and let ω be a subset of Ω. Suppose thatwe wish to test

H0 : θ ∈ ω against Ha : θ ∈ Ω− ω. (1)

Let us denote the nonempty subspaces of Ω by ω1, ω2, ..., ωk with the ordering

Ω = ω1 ⊃ ω2 ⊃ ω3 ⊃ ... ⊃ ωk = ω. (2)

We then test the hypotheses

H0(i) : θ ∈ ωi against Ha(i) : θ ∈ Ω− ωi, i = 2, ..., k. (3)

The hypotheses H0(i) and Ha(i), i = 2, ..., k, are called nested hypotheses.Let Si denote the test statistic for testing

H0(i) : θ ∈ ωi against Ha(i) : θ ∈ Ω− ωi. (4)

Given the independence of S2, ..., Sk, the nested hypotheses are tested iteratively.A test statistic Si rejecting H0(i) at a prespecified significance level α is regarded assufficient evidence against H0. If all the test statistics Si, i = 2, .., k do not reject

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the null hypotheses H0(i), i = 2, ..., k, then the p-value of the test, denoted by P , iscalculated as

P = 1−k∏

i=2

(1− Pi), (5)

where Pi is the p-value for Si.The concept of nested hypotheses can be seen as an application of the union-

intersection principle first introduced by Roy (1953) and discussed in detail byOlkin and Tomsky (1981). The union-intersection principle is a heuristic methodof test construction. Let ωr, ω

∗r , r ∈ Γ be a collection of sets in the parameter

space of the parameter θ where Γ is an arbitrary index set. Assume that∩r∈Γ

ωr is

nonempty. Define component hypotheses H0(r) and Ha(r) as H0(r) : θ ∈ ωr andHa(r) : θ ∈ ω∗

r . The original null hypothesis H0 is accepted if and only if each ofthe component hypotheses H0(r), r ∈ Γ is accepted. If at least one of the H0(r) isrejected, then H0 is rejected.

McDermott and Mudholkar (1993) consider different ways of combining testsfrom finite component hypotheses. In particular, McDermott and Mudholkar (1993)consider four different ways of combining independent p-values Pi, i = 2, ..., kwhich correspond to the test statistics Si, i = 2, ..., k arising from nested hypothe-ses. The four p-value combining methods are summarized below:

1. Tippett’s (1931) combination

ΨT = min(P2, ..., Pk) (6)

which follows a min(U2, ..., Uk) distribution under H0, where U2, ..., Uk areindependent and Ui ∼ U(0, 1), i = 2, ..., k,

2. Fisher’s (1932) combination

ΨF = −2k∑

i=2

logPi (7)

which follows a χ22k−2 distribution under H0,

3. Liptak’s (1958) combination

ΨN =k∑

i=2

Φ−1(1− Pi) (8)

which follows a N(0, k − 1) distribution under H0,

4. Mudholkar-George (1979) combination

ΨL = −[π2(k − 1)(5k − 3)

3(5k − 1)

]−1/2 k∑i=2

log

(Pi

1− Pi

)(9)

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which has a distribution that is well approximated by a Student’s t distribu-tion with 5k − 1 degrees of freedom under H0.

Small values of ΨT and large values of ΨF , ΨN , and ΨL are seen as evidenceagainst H0. Mudholkar et al. (1995) call this approach a “finite-intersection ap-proach” in a broader sense allowing for different ways of combining tests fromsuch finite component hypotheses.

In practice, the null hypothesis H0 and its nested hypotheses H0(i), i = 2, ..., koften take the form for a parameter of interest θi,i = 1, ..., k:

H0 : θ1 = θ2 = ... = θk = θ, (10)H0(i) : [θ1, θ2, ..., θi−1] = θi = θ. (11)

In (11), the parameters grouped in the brackets [θ1, θ2, ..., θi−1] lack any orderrestrictions. The samples whose parameters are in the brackets are combined andregarded as one sample. The ordered alternatives in an increasing order and theirnested hypotheses that Hogg (1962) and McDermott and Mudholkar (1993) con-sider are expressed as

Ha : θ1 ≤ θ2 ≤ ... ≤ θk, with at least one strict inequality, (12)Ha(i) : [θ1, θ2, ..., θi−1] < θi. (13)

Note the possibility of considering the ordered alternatives in a decreasing or-der. The formulation of such hypotheses is symmetric to the formulation of anincreasing order.

3 New Test for Equality of Variances against Ordered Alternatives

Let Xi1, ..., Xini, i = 1, ..., k represent k independent samples, where in each sam-

ple random variables Xij, j = 1, ..., ni, are independent and identically distributed(i.i.d.) with a continuous distribution having a finite second moment mi. Let usdenote the distribution function by Gi(xij) = G ((xij − µi)/σi) where µi and σi rep-resent the mean and standard deviation of the ith sample respectively. We focuson tests for equality of variances against ordered alternatives in the presence ofunknown and possibly unequal location:

H0 : σ1 = σ2 = ... = σk = σ, (14)Ha : σ1 ≤ σ2 ≤ ... ≤ σk with at least one strict inequality, (15)

The absolute deviation of Xij from the ith population mean µi and the samplemean Xi are denoted by Yij = |Xij − µi| and Zij = |Xij − Xi| respectively. By thefinite second moment assumption of G, under the assumptions of H0, we can writeE(Yij) = τ and Var(Yij) = η2 < ∞. Note that τ = dσ where

d =2(µiGi(µi)−Hi(µi))√

mi − µ2i

, Hi(µi) =

∫ µi

−∞t dGi.

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Further, assume that ni/N := λi → ci ∈ (0, 1), a nonzero constant less than 1,

where N =k∑

i=1

ni. In addition, let us introduce the notations n(i−1) =i−1∑m=1

nm and

λ(i−1) =i−1∑m=1

λm, i = 2, ..., k.

The approach taken in this article for constructing such a test relies on the k− 1nested hypotheses H0(i), i = 2, ..., k, which are subsets of the original hypothesisdenoted as follows:

H0(i) : [σ1, σ2, ..., σi−1] = σi = σ, (16)Ha(i) : [σ1, σ2, ..., σi−1] < σi. (17)

Let Di = n−1i

ni∑j=1

Zij and D(i−1) = n−1(i−1)

i−1∑m=1

nmDm, i = 2, ..., k. Using the idea

of the usual two-sample t-test considered by Hogg (1962) in the cases of unequalsample sizes, we can construct a test statistic for H0(i), denoted by Ti, i = 2, ..., k,by comparing Di and D(i−1) as follows:

Ti =

√N(Di −D(i−1))

spi√

1/λ(i−1) + 1/λi

(18)

where

spi =

√(n(i−1) − 1)s2(i−1) + (ni − 1)s2i

n(i−1) + ni − 2, (19)

s2i =

ni∑j=1

(Zij −Di)2

ni − 1,

s2(i−1) =

i−1∑m=1

nm∑j=1

(Zmj −D(i−1))2

n(i−1) − 1.

McDermott and Mudholkar (1993) suggest that the p-values associated withstatistics from the k − 1 nested hypotheses (16), P2, ..., Pk, are combined to test H0

using one of ΨT , ΨF , ΨN , and ΨL (6, 7, 8, and 9). However, the four p-value com-bining methods rely on the assumption that the p-values come from independentstatistics. Nevertheless, asymptotic independence of T2, ..., Tk, stated below, es-tablishes good approximations to the true p-value for the H0 using such p-valuecombining methods.

We have the following results on the asymptotic properties of the test statisticTi under the assumptions of H0.

Theorem 1 As min(n1, ..., nk) → ∞, TiD−→ N(0, 1), i = 2, ..., k.

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Theorem 2 As min(n1, ..., nk) → ∞, (T2, ..., Tk)D−→ Nk−1(0, I), where 0 denotes the

zero column vector with k− 1 components, and I is the (k− 1)× (k− 1) identity matrix.

From Theorem 2, we immediately have the following corollary.

Corollary 1 As min(n1, ..., nk) → ∞, T2, ..., Tk are asymptotically independent.

Proofs of the Theorems are based on two auxiliary lemmas (See Lemma 1 and 2 inthe Appendix).

Remark 1 Levene’s transformation Zij = |Xij − Xi| can be replaced with Brown-Forsythe transformation Z50ij = |Xij−X50i| where X50i denotes the sample median.Conover et al. (1981) and Lim and Loh (1996) recommend the Brown-Forsythemodification of the Levene’s test (BFL) as a test for equality of variances as it isrobust to nonnormality and has competitive powers. Carroll and Schneider (1985)give a theoretical justification for the asymptotically correct level of test for BFL,which is also applicable to the new test. Theorem 1 and 2 hold for the statisticswith Brown-Forsythe modification.

Remark 2 As Hogg (1962) notes, the new test can be used to test equality of vari-ances against general alternatives σi = σj, i = j, using two-sided tests instead ofone-sided for ordered alternatives.

4 Modifications for Unequal or Small Sample Sizes

In the real-life situations, we often encounter experiments with unbalanced de-signs: i.e. unequal number of observations per sample. An unbalanced designmay be caused due to problems in the data collection, such as unexpected deathsof specimens, resulting in a loss of observations in what would otherwise havebeen a balanced design (Cabrera and McDougall, 2002), or intentional in order toreflect the population proportion of the samples (Clark-Carter, 1997).

Keyes and Levy (1997) show that the size and power of Levene’s test are af-fected by unbalanced designs with small sample sizes, which is also of concern toour new test (18). The new test can be made more robust using a correction factorfor unequal sample sizes (O’Brien, 1978; Keyes and Levy, 1997) or the Hines-Hines(2000) structural zero removal method. In this section, we introduce a new proce-dure for combining the correction factor 1/

√1− 1/ni (O’Brien, 1978) and a slight

modification of the structural zero removal method (Hines and Hines, 2000).For a sample with an even sample size, consider a transformation of Z50i(1) and

Z50i(2) into Z50i(1) − Z50i(2)(= 0) and Z50i(1) + Z50i(2)(= 2Z50i(1)) respectively. Then,the newly created structural zero (Z50i(1) − Z50i(2)(= 0)) is removed. For a samplewith an odd sample size, perform the original Hines-Hines structural zero method.

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Let D50i and D50i(NS) be the sample means of Z50ij before and after the proposedstructural zero removal respectively. Then, these two quantities are related by

D50i =

(1− 1

ni

)D50i(NS). (20)

Therefore, for a one-way ANOVA design where the samples are from standardnormal populations,

E(D50i(NS)) =

(κni

1− 1/ni

)σi (21)

where κniis a constant that depends on ni (O’Neil and Mathews, 2000). Also,

κni/√2/π is well approximated by

√1− 1/ni. Following the idea of Keyes and

Levy (1997), we introduce a new correction factor for Z50ij after the modified struc-tural zero removal,

ϕni=

√1− 1

ni

, (22)

so that

ϕniE(D50i(NS)) =

(κni√

1− 1/ni

)σi ≈

√2

πσi. (23)

Note that the correction factor ϕnishould be applied to each observation firsthand

(prior to the calculation of test statistics) based on the original sample size ni inorder to estimate the standard deviation more accurately.

Remark 3 we show that the correction factor ϕnimakes our new test more robust

under both normal and non-normal distributions through simulations in Section 5even though ϕni

is derived under the normality assumption of samples.

5 Simulation Plan

The size and power of the tests are investigated in the simulation study using fourdistributions and five combinations of the sample sizes for k = 3 and k = 6. Forestimating the size, nominal 5% level is used throughout the study with 10,000Monte Carlo simulations. Calculations of the power estimates are based on thesimulated exact critical values at the 5% level with 10,000 Monte Carlo simula-tions. The distributions used are: 1. normal, 2. exponential, 3. Laplace, and 4.NIG(7,2), where NIG(a,b) is the normal inverse Gaussian distribution (Atkinson1982; Barndorff-Nielsen and Blaesild, 1983) with excess kurtosis a, skewness b,and mean 0 . 1 The choice of distributions covers a broad range of characteristics

1Note that normal inverse Gaussian distribution is typically characterized by four parameters α′

(tail heaviness), β′

(asymmetry), δ′

(scale), and µ′

(location). However, given the mean λ, variance

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including symmetric to skewed and light- to heavy-tailed. Furthermore, differentsample sizes, both equal and unequal, are considered in this study.

We have four different tests for the simulation study. They are: 1. LN: New testpresented (18) based on Hogg’s test (1962) with Levene’s transformation (Lnestedtest); 2. LT: Ltrend test (Hui et al., 2008); 3. DCT: Double contrast test with Levene’stransformation (Neuhauser and Hothorn, 2000); 4. M95: Modified McDermott-Mudholkar test with Miller’s pseudovalues (Mudholkar et al., 1995). We also have:5. L: Levene’s test (Levene, 1960) as a comparison to the four trend tests.

The first three tests (LN, LT, DCT) and L are named Levene-type tests as theyutilize Levene’s transformation. For the Levene-type tests, Brown-Forsythe modi-fication with the combination of modified structural zero removal and the correc-tion factor ϕni

are considered. Mean-based and Brown-Forsythe modification testswithout the correction factor are also investigated, and found to be less robust ingeneral. For the tests with nested hypotheses (LN and M95), the size and powerusing ΨN (8) and ΨL (9) are shown as these two methods tend to produce betterestimates of the size and competitive power for small sample sizes. Levene-typetests with the correction factor ϕni

are denoted by TEST(ϕni).

From the Tables 1 and 2, we see that LN(ϕni) and LT(ϕni

) are very robust underskewed or heavy-tailed distributions and different sample sizes in general. How-ever, DCT(ϕni

) tends to underestimate the size under skewed distributions withunequal sample sizes, and M95 tends to overestimate the size under skewed dis-tributions.

By looking at the power comparisons (Tables 3–6), LN(ϕni) is often the most

powerful across different distributions and sample sizes compared to the othertests. In particular, LN(ϕni

) is the most advantageous in the situations where di-minishingly increasing standard deviation patterns are observed. When there isa linear standard deviation pattern, LT(ϕni

) is very powerful, but LN(ϕni) is also

competitive. DCT(ϕni) is less powerful compared to LN(ϕni

) in general. M95 isalso powerful, but is less competitive than LN(ϕni

) under heavy-tailed or skeweddistributions.

Based on the simulation study, we find that LN(ϕni) is robust and very pow-

erful under different distributions, sample sizes, and different standard deviationpatterns. When there is a linear standard deviation pattern is suspected, LT(ϕni

)may be used as an alternative. DCT(ϕni

) and M95 are fairly robust and powerful,but are less competitive under heavy-tailed or skewed distributions.

σ2, excess kurtosis κ, and skewness γ with 3κ > 5γ2, we can calculate the four parameters α′, β

′, δ

′,

and µ′

by

α′=

3√3κ− 4γ2

σ(3κ− 5γ2), β

′=

3γ

σ(3κ− 5γ2), δ

′=

σ2√(α′2 − β′2)3

α′2, µ

′= λ− δ

′β

′√α′2 − β′2

. (24)

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6 Case Studies

We present several cases where our new test for detecting a trend in variances isuseful. In this section, all the Levene-type tests (LN, LT, DCT, L) are based onthe Brown-Forsythe modification with the correction factor ϕni

applied after thestructural zero removal. We use ΦN to calculate the p-values of the tests with nestedhypotheses (LN, M95).

6.1 Monthly Apartment Rent in Thalwil, Switzerland

The first example is the October 2008 monthly rent data (N = 25) for apartmentsin Thalwil, Switzerland. We are interested in detecting a trend in variability asan increase in variability in rent would imply a wider spread in the quality orcondition within the group of apartments. The observations are grouped accordingto the number of rooms available in the apartment: 1. Small (less than 3 rooms)(n1 = 5); 2. Medium (3 or 3.5 rooms) (n2 = 8); and 3. Large (more than 3.5 rooms)(n3 = 12). We conjecture an increasing pattern in variability in the monthly rent asthe apartment size increases. Since the sample sizes are small, it is difficult to judgewhether the samples are normally distributed. Therefore, robust tests for non-normal samples are more favorable in order to assess the trend more accurately.Among the Levene-type tests, LN has the lowest p-value of 0.0171. The p-valuesprovided by LT and DCT, 0.0553 and 0.0876 respectively, are on the boarder ofsignificance. M95 has the p-values of 8.18×10−3. However, M95 tends to overrejectthe null hypothesis for non-normal samples. Therefore, we can conclude that thereis some evidence of an increasing trend in variability based on LN with the p-valueof 0.0171. On the other hand, L provides the p-value of 0.289, not rejecting theequality in variability among the samples. The phenomenon may be attributed toa weak power of Levene’s test when the sample sizes are small and have a non-decreasing pattern.

6.2 Temperature-accelerated Life Test of a Type of Sheathed Tabular Heater

The second example is taken from the temperature-accelerated life test data of atype of sheathed tabular heater studied by Nelson (1972). At each of four temper-atures (1708, 1660, 1620, and 1520F ), 6 heaters were tested and the numbers ofhours to failure were recorded (i.e. ni = 6, i = 1, 2, 3, 4, and N = 24). An increasein variability would imply more uncertainty in life of the heater, and is of interestfrom the quality control aspects. We conjecture an increasing pattern in variabilitywith a decrease in temperature. To determine whether there is an increasing trendin variability, robust tests for non-normal samples are preferred in this example aswell since the sample sizes are small. For LT, linear scores (1, 2, 3, 4) and non-linearscores (−1708,−1660,−1620,−1520) are applied. Among all the tests we have, LNhas the lowest p-value of 1.78 × 10−11, followed by LT having the non-linear scoreand DCT with 1.44 × 10−9 and 2.92 × 10−9 respectively. Therefore, there is strongevidence of an increasing trend in variability with a decrease in temperature. L

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provides the p-value of 4.79× 10−8, strongly rejecting the null hypothesis of equalvariability.

6.3 Soil Lead Concentration in Syracuse, New York

The next example refers to the soil lead concentration study in Syracuse, New York(Griffith (2002)). A high level of soil lead concentration is hazardous to humanbody, and thus we are interested in investingating the distributions of the soillead concentration in different locations. The soil samples (N = 167) are classi-fied by their location features as follows: 1. park soil (n1 = 30); 2. playground soil(n2 = 17); 3. streetside soil (n3 = 74); 4. house lot soil (n4 = 30); and 5. vacantlot soil (n5 = 16). Griffith (2002) lists major sources of soil lead contamination as 1.widespread use of lead-based paints; 2. lead emissions in gasoline in earlier years;and 3. lead waste from mining/commercial/manufacturing processes. Consider-ing the exposures to these sources, we conjecture the order of the variabilities inthe soil lead concentration among the five location features as 1 < 2 < ... < 5 forthe following reasoning.

• Park and playground soil is located away from these sources of pollution.

• Playgrounds tend to be located closer to the sources of contamination com-pared to parks.

• Streetside soil would probably have a comparatively lower variability due toits limited area with relatively uniform influences from these three sources ofpollution.

• House lot soil would have a slightly higher variability due to paints. Presenceof roofs and walls would lower the variability.

• Vacant lot soil would have the highest variability because of absence of roofsand walls, and proximity to the sources of pollution.

We observe that there is some evidence of non-normality in the sample distri-butions. For example, the streetside and house lot samples have the p-values lessthan 0.01 for the Shapiro-Wilk test. Therefore, robust tests for non-normal samplesare preferred in order to assess the trend more accurately. Among the Levene-typetests and M95, LN has the lowest p-value of 3.03 × 10−9, followed by DCT and LTwith the p-values of 5.17 × 10−9 and 8.66 × 10−9 respectively. Therefore, there isstrong evidence of an increasing trend in variability which supports the conjecturebased on the tests above. L provides the p-value of 4.16×10−8, suggesting unequalvariability among samples.

6.4 Testosterone Levels with Different Smoking Habits

The last example explores the differences in variability of the testosterone levelsof men with different smoking habits. Testosterone is the most important sex hor-

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mone that men have, and low testosterone is known to cause various health prob-lems (Cunningham et al., 2004). Finding a trend in variability in the testosteronelevels according to smoking habits is a study of interest to endocrinologists. Le(1994) classifies four groups of men, 10 in each group, according to their smokinghabits: 1. heavy smokers; 2. light smokers; 3. former smokers; and 4. non-smokers(i.e. ni = 10, i = 1, 2, 3, 4, and N = 40). We conjecture an increasing pattern invariability with a decrease in exposure to smoking. To determine whether thereis an increasing trend, we prefer robust tests for non-normality in this example asthe sample sizes are relatively small. For LT, linear scores (1, 2, 3, 4) are assignedto each group. All the tests reject the null hypothesis strongly, indicating an in-creasing trend in variability with a decrease in exposure to smoking. M95 hasthe lowest p-value of 6.76 × 10−5, followed by LN and LT with 1.85 × 10−4 and1.00×10−3 respectively. However, M95 tends to have a slightly inflated size, so thep-value should be interpreted with caution. In contrast, L provides the p-value of0.0191, marginally rejecting the null hypothesis of equal variability among groups.

7 Conclusions and Future Research

The purpose of this article is to present a robust test for equality of variancesagainst ordered alternatives for k independent samples and to show its theoret-ical reasoning. The test is developed by combining Levene’s transformation andthe finite-intersection approach. Levene’s transformation stabilizes the size of thetest for non-normal distributions while the finite-intersection method increases thepower of the test. The theoretical reasoning is given by proving asymptotic nor-mality and independence of the component statistics Ti, i = 2, 3, ..., k. To make thetest more robust for unequal or small sample sizes, a new correction factor ϕni

forZ50ij after the modified structural zero removal is considered. The Monte Carlosimulation study suggests that the new test with such modification (LN(ϕni

)) pre-serves the size very well and is powerful compared to the existing tests for equalityof variances against ordered alternatives for different sample sizes under normal,skewed, and heavy-tailed distributions. The new test is applicable to a wide va-riety of disciplines such as microeconomics, engineering, environmental studies,and health studies.

In the future, we plan to investigate an application of the bootstrap method(Boos and Brownie, 1989; Lim and Loh, 1996) to our new test, which is expected toincrease the robustness and power of the test further, or consider different ways todecompose H0 and Ha into nested hypotheses by considering more general casesof grouping the samples, such as those considered by McDermott and Mudholkar(1993).

Acknowledgement

We would like to thank Drs. David Johnson, Daniel Griffith, and Jennifer Bretschfor providing the soil lead concentration data in Syracuse, New York, and Adrian

12

Waddell for providing the apartment rent data in Thalwil, Switzerland. We are alsovery thankful to Professors Jeannette O’Hara-Hines and Joel Dubin for stimulatingdiscussions and advises. The research was supported by a grant from the NationalScience and Engineering Research Council of Canada. This work was made pos-sible by the facilities of the Shared Hierarchical Academic Research ComputingNetwork (SHARCNET: http://www.sharcnet.ca).

8 Appendix A: Lemmas for the Theorems

Lemma 1 gives convergence in probability of random variables of interest, fol-lowed by a proof of Theorem 1 which provides the asymptotic normality of Ti.Lemma 2 establishes conditions necessary to use the Cramer-Wold device (Huand Rosenberger, 2006) which is employed in the proof of Theorem 2 to provethe asymptotic multivariate normality of the joint distribution of (T2, ..., Tk) withthe (k − 1) × (k − 1) identity matrix as the variance-covariance matrix. All thelemmas and theorems are derived under the assumptions of H0.

Let n(i−1) =i−1∑m=1

nm, Di = n−1i

ni∑j=1

Zij , and D(i−1) = n−1(i−1)

i−1∑m=1

nmDm, i = 2, ..., k.

Also, let s2pi be as defined in (19). Then, we have the following.

Lemma 1 As min(n1, ..., ni) → ∞, Dip−→ τ , D(i−1)

p−→ τ , and s2pip−→ η2.

Proof. Gastwirth (1982) shows Dip−→ τ . Also,

D(i−1) =1

n(i−1)

i−1∑m=1

nmDmp−→ 1

n(i−1)

i−1∑m=1

nmτ = τ.

Similar arguments using the results Dip−→ τ and D(i−1)

p−→ τ yield s2ip−→ η2 and

s2(i−1)

p−→ η2. Therefore,

s2pip−→

(n(i−1) − 1)η2 + (ni − 1)η2

n(i−1) + ni − 2= η2.

Proof of Theorem 1. Let us consider the test statistic

Ti =

√N(Di −D(i−1))

spi√1/λ(i−1) + 1/λi

.

Denote

Qi =

√N(Di −D(i−1))

η√1/λ(i−1) + 1/λi

.

13

Then, Ti = ηs−1pi Qi. By Lemma 1, ηs−1

pi

p−→ 1. Consider the convergence of randomvariable Qi. Let

Wm =

√nm(Dm − τ)

η,m = 1, ..., i.

Then,

Qi =1√

1/λ(i−1) + 1/λi

[1√λi

Wi −i−1∑m=1

λm

λ(i−1)

1√λm

Wm

].

Gastwirth (1982) shows that WmD−→ N(0, 1) for m = 1, ..., i. Also, by the as-

sumption of k independent samples, D1, ..., Dk are independent. Thus, W1, ...,Wk

are also independent. Therefore, QiD−→ N(0, ξ2i ), and by the Slutsky’s Theorem

(Lehmann, 1998), TiD−→ N(0, ξ2i ) for some ξ2i > 0, i = 1, ..., k. Now, let us calculate

the value of ξ2i . We have λm → cm ∈ (0, 1) for m = 1, ..., i. Hence,

ξ2i =

(1√

1/c(i−1) + 1/ci

)2 [(1

√ci

)2

+i−1∑m=1

(cm

c(i−1)

1√cm

)2]= 1,

where c(i−1) :=i−1∑m=1

cm. This completes the proof.

Lemma 2 proves conditions necessary to use the Cramer-Wold device (Hu andRosenberger, 2006), which is used for proving Theorem 2.

Lemma 2 As min(n1, ..., nk) → ∞,k∑

i=2

biQiD−→ N

(0,

k∑i=2

b2i

)for any

bi ∈ ℜ, i = 2, ..., k.

Proof. We shall prove this result by induction. For the base cases, consider k = 2

and k = 3. When k = 2, b2Q2D−→ N (0, b22) by Theorem 1.When k = 3, a straightfor-

ward calculation shows that3∑

i=2

biQiD−→ N

(0,

3∑i=2

b2i

). For the induction hypoth-

esis, assume that, for k = 2, ..., r − 1, r ≥ 4,k∑

i=2

biQiD−→ N

(0,

k∑i=2

b2i

)holds. The

induction hypothesis has the following implications. Routine calculations showthat

k∑i=2

biQi =k∑

i=1

fk,i(λ1, ..., λk)Wi,

14

where

fk,1(λ1, ..., λk) = −√λ1

(k∑

i=2

bi

λ(i−1)

√1/λ(i−1) + 1/λi

),

fk,i(λ1, ..., λk) =1√λi

(λi

k−1∑m=i

bm+1

λ(m)

√1/λ(m) + 1/λm+1

)− bi√

1/λ(i−1) + 1/λi

,

i = 2, ..., r − 1,

fk,k(λ1, ..., λk) =1√λk

bk√1/λ(k−1) + 1/λk

,

for any k ≥ 3.As min(n1, ..., nk) → ∞, fk,i(λ1, ..., λk) → fk,i(c1, ..., ck) for i = 1, ..., k. Also, by

Theorem 1, W1, ...,Wr are independent and WiD−→ N(0, 1) for i = 1, ..., r. There-

fore,r∑

i=2

biQiD−→ N

(0,

r∑i=1

fr,i(c1, ..., cr)2)

. Thus, the induction hypothesis im-

plies thatk∑

i=1

fk,i(c1, ..., ck)2 =k∑

i=2

b2i for k = 3, ..., r − 1.

Consider k = r. The calculations above show thatr∑

i=2

biQi =r∑

i=1

fr,i(λ1, ..., λr)Wi.

Following the argument above, we have,r∑

i=2

biQiD−→ N

(0,

r∑i=1

fr,i(c1, ..., cr)2)

.

Moreover,r∑

i=1

fr,i(c1, ..., cr)2 = gr,1(c1, ..., cr) + gr,2(c1, ..., cr) +r−1∑i=1

fr−1,i(c1, ..., cr−1)2,

where

gr,1(c1, ..., cr) =b2r

1/c(r−1) + 1/cr

[1

cr+

r−1∑i=1

(ci

cr−1

)2(1

ci

)],

gr,2(c1, ..., cr) = 2

(br

c(r−1)

√1/c(r−1) + 1/cr

)[r−1∑i=2

bi√1/c(i−1) + 1/ci

(−1 +

1

c(i−1)

i−1∑m=1

cm

)].

Routine calculations show that gr,1(c1, ..., cr) = b2r and gr,2(c1, ..., cr) = 0. Also, by

the induction hypothesis,r−1∑i=1

fr−1,i(c1, ..., cr−1)2 =r−1∑i=2

b2i . Therefore,r∑

i=1

fr,i(c1, ..., cr)2 =

b2r +r−1∑i=2

b2i =r∑

i=2

b2i . Hence,

r∑i=2

biQiD−→ N

(0,

r∑i=2

b2i

).

15

This completes the induction.

Proof of Theorem 2. By the Cramer-Wold device (Corollary A.3. of Hu and Rosen-

berger (2006)), (T2, ..., Tk)D−→ Nk−1(0, I) if and only if

k∑i=2

biTiD−→ N

(0,

r∑i=2

b2i

)for

any nonzero b2, ..., bk ∈ ℜ. We shall prove the latter condition by using Lemma 2.The identity Ti = Qi +

(ηs−1

pi − 1)Qi gives the following;

k∑i=2

biTi =k∑

i=2

bi[Qi +

(ηs−1

pi − 1)Qi

]=

k∑i=2

biQi +k∑

i=2

bi(ηs−1

pi − 1)Qi.

By Lemma 2,k∑

i=2

biQiD−→ N

(0,

k∑i=2

b2i

), and by Lemma 1, ηs−1

pi − 1p−→ 0. The

random variable Qi converges in distribution by Theorem 1. Thus, Qi is boundedin probability.

Therefore,(ηs−1

pi − 1)Qi

p−→ 0. Hence,k∑

i=2

bi(ηs−1

pi − 1)Qi

p−→ 0. By the Slutsky’s

Theorem (Lehmann, 1998),k∑

i=2

biTiD−→ N

(0,

k∑i=2

b2i

)for any b2, ..., bk ∈ ℜ. Hence,

by the Cramer-Wold device, (T2, ..., Tk)D−→ Nk−1(0, I).

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20

Normal Distribution(σ1, σ2, σ3, σ4) = (1, 1, 1, 1)

Sample Size L(ϕni) LT(ϕni

) LN(ΨN ,ϕni) LN(ΨL,ϕni

) DCT(ϕni) M95(ΨN ) M95(ΨL)

(5,5,5,5) 0.0338 0.0439 0.0489 0.0507 0.0406 0.0390 0.0360(10,10,10,10) 0.0575 0.0500 0.0539 0.0551 0.0497 0.0467 0.0443(20,20,20,20) 0.0493 0.0482 0.0507 0.0532 0.0476 0.0511 0.0483(5,10,10,20) 0.0458 0.0607 0.0630 0.0638 0.0496 0.0610 0.0629(20,10,10,5) 0.0447 0.0408 0.0432 0.0468 0.0413 0.0487 0.0466

Exponential Distribution(σ1, σ2, σ3, σ4) = (1, 1, 1, 1)




(5,5,5,5) 0.0479 0.0502 0.0615 0.0647 0.0464 0.0642 0.0683(10,10,10,10) 0.0667 0.0577 0.0672 0.0705 0.0582 0.0747 0.0785(20,20,20,20) 0.0535 0.0550 0.0598 0.0622 0.0540 0.0738 0.0779(5,10,10,20) 0.0559 0.0526 0.0679 0.0687 0.0332 0.0878 0.0914(20,10,10,5) 0.0622 0.0527 0.0593 0.0654 0.0630 0.0792 0.0870

Laplace Distribution(σ1, σ2, σ3, σ4) = (1, 1, 1, 1)




(5,5,5,5) 0.0426 0.0495 0.0548 0.0572 0.0469 0.0512 0.0509(10,10,10,10) 0.0628 0.0608 0.0644 0.0665 0.0598 0.0639 0.0652(20,20,20,20) 0.0501 0.0524 0.0528 0.0538 0.0500 0.0561 0.0571(5,10,10,20) 0.0551 0.0497 0.0556 0.0555 0.0308 0.0731 0.0749(20,10,10,5) 0.0554 0.0510 0.0536 0.0603 0.0571 0.0597 0.0634

Normal Inverse Gaussian Distribution with Excess Kurtosis 7, Skewness 2(σ1, σ2, σ3, σ4) = (1, 1, 1, 1)




(5,5,5,5) 0.0471 0.0462 0.0536 0.0560 0.0422 0.0558 0.0581(10,10,10,10) 0.0597 0.0566 0.0627 0.0657 0.0533 0.0712 0.0778(20,20,20,20) 0.0507 0.0537 0.0583 0.0606 0.0502 0.0733 0.0816(5,10,10,20) 0.0569 0.0434 0.0619 0.0602 0.0261 0.0757 0.0809(20,10,10,5) 0.0634 0.0549 0.0581 0.0655 0.0650 0.0791 0.0879

Table 1: Estimated size of the tests for k = 4.

Normal Distribution(σ1, σ2, σ3, σ4, σ5, σ6) = (1, 1, 1, 1, 1, 1)




(5,5,5,5,5,5) 0.0304 0.0400 0.0472 0.0486 0.0375 0.0374 0.0349(10,10,10,10,10,10) 0.0580 0.0505 0.0527 0.0560 0.0491 0.0414 0.0393(20,20,20,20,20,20) 0.0482 0.0556 0.0542 0.0567 0.0530 0.0497 0.0482

(5,5,10,10,20,20) 0.0457 0.0612 0.0666 0.0683 0.0496 0.0587 0.0591(20,20,10,10,5,5) 0.0453 0.0319 0.0361 0.0401 0.0367 0.0557 0.0510

Exponential Distribution(σ1, σ2, σ3, σ4, σ5, σ6) = (1, 1, 1, 1, 1, 1)




(5,5,5,5,5,5) 0.0464 0.0470 0.0609 0.0682 0.0470 0.0570 0.0601(10,10,10,10,10,10) 0.0692 0.0530 0.0619 0.0670 0.0548 0.0649 0.0692(20,20,20,20,20,20) 0.0522 0.0522 0.0589 0.0630 0.0542 0.0662 0.0704

(5,5,10,10,20,20) 0.0565 0.0516 0.0688 0.0716 0.0315 0.0801 0.0855(20,20,10,10,5,5) 0.0532 0.0459 0.0504 0.0595 0.0548 0.0729 0.0838

Laplace Distribution(σ1, σ2, σ3, σ4, σ5, σ6) = (1, 1, 1, 1, 1, 1)




(5,5,5,5,5,5) 0.0473 0.0531 0.0603 0.0653 0.0488 0.0546 0.0531(10,10,10,10,10,10) 0.0622 0.0547 0.0601 0.0661 0.0555 0.0610 0.0612(20,20,20,20,20,20) 0.0529 0.0521 0.0525 0.0551 0.0515 0.0540 0.0560

(5,5,10,10,20,20) 0.0541 0.0505 0.0643 0.0638 0.0312 0.0733 0.0752(20,20,10,10,5,5) 0.0586 0.0515 0.0527 0.0620 0.0608 0.0656 0.0708

Normal Inverse Gaussian Distribution with Excess Kurtosis 7, Skewness 2(σ1, σ2, σ3, σ4, σ5, σ6) = (1, 1, 1, 1, 1, 1)




(5,5,5,5,5,5) 0.0448 0.0458 0.0549 0.0621 0.0477 0.0539 0.0588(10,10,10,10,10,10) 0.0642 0.0551 0.0619 0.0697 0.0532 0.0684 0.0768(20,20,20,20,20,20) 0.0513 0.0513 0.0577 0.0603 0.0509 0.0674 0.0750

(5,5,10,10,20,20) 0.0568 0.0514 0.0707 0.0719 0.0335 0.0759 0.0805(20,20,10,10,5,5) 0.0588 0.0504 0.0559 0.0650 0.0628 0.0761 0.0876

Table 2: Estimated size of the tests for k = 6.

21

Normal Distribution(σ1, σ2, σ3, σ4) = (1, 2, 2, 2)




(5,5,5,5) 0.1139 0.2253 0.2849 0.2821 0.2201 0.3145 0.3176(10,10,10,10) 0.2658 0.4145 0.5752 0.5837 0.4962 0.6688 0.6735(20,20,20,20) 0.6630 0.6674 0.8877 0.9107 0.8526 0.9416 0.9493(5,10,10,20) 0.1045 0.1830 0.2859 0.2706 0.2752 0.3513 0.3532(20,10,10,5) 0.5868 0.7032 0.8232 0.8344 0.6045 0.8343 0.8444

Exponential Distribution(σ1, σ2, σ3, σ4) = (1, 2, 2, 2)




(5,5,5,5) 0.0765 0.1494 0.1876 0.1785 0.1244 0.2090 0.2049(10,10,10,10) 0.1281 0.2029 0.2939 0.2864 0.2161 0.2900 0.2888(20,20,20,20) 0.2834 0.4302 0.5792 0.5879 0.4949 0.5123 0.5060(5,10,10,20) 0.0536 0.1223 0.1737 0.1710 0.1559 0.1948 0.1913(20,10,10,5) 0.3028 0.4017 0.4927 0.4869 0.3126 0.4282 0.4314

Laplace Distribution(σ1, σ2, σ3, σ4) = (1, 2, 2, 2)




(5,5,5,5) 0.0846 0.1524 0.1933 0.1886 0.1408 0.2298 0.2345(10,10,10,10) 0.1628 0.2804 0.3948 0.3885 0.2944 0.3930 0.3924(20,20,20,20) 0.4010 0.4761 0.6738 0.6927 0.6060 0.6595 0.6690(5,10,10,20) 0.0604 0.1346 0.1957 0.1881 0.1921 0.2329 0.2340(20,10,10,5) 0.4043 0.4762 0.5866 0.5912 0.3692 0.5264 0.5277

Normal Inverse Gaussian Distribution with Excess Kurtosis 7, Skewness 2(σ1, σ2, σ3, σ4) = (1, 2, 2, 2)




(5,5,5,5) 0.0801 0.1439 0.1976 0.1897 0.1402 0.2249 0.2246(10,10,10,10) 0.1376 0.2492 0.3518 0.3451 0.2527 0.3549 0.3608(20,20,20,20) 0.3061 0.4206 0.6027 0.6212 0.5395 0.5321 0.5347(5,10,10,20) 0.0498 0.1277 0.1912 0.1923 0.1734 0.2373 0.2309(20,10,10,5) 0.3505 0.4236 0.5140 0.5144 0.3071 0.4250 0.4227

Table 3: Estimated power of the tests for k = 4 with empirical critical values.

Normal Distribution(σ1, σ2, σ3, σ4) = (log 2, log 3, log 4, log 5)




(5,5,5,5) 0.1393 0.4042 0.4225 0.4196 0.3488 0.3787 0.3786(10,10,10,10) 0.3493 0.7092 0.7621 0.7588 0.6743 0.7604 0.7528(20,20,20,20) 0.7744 0.9455 0.9760 0.9759 0.9466 0.9789 0.9776(5,10,10,20) 0.1972 0.5571 0.6209 0.6035 0.5367 0.5260 0.5191(20,10,10,5) 0.5785 0.8104 0.8468 0.8461 0.7245 0.8417 0.8465

Exponential Distribution(σ1, σ2, σ3, σ4) = (log 2, log 3, log 4, log 5)




(5,5,5,5) 0.0870 0.2498 0.2586 0.2528 0.2054 0.2449 0.2419(10,10,10,10) 0.1624 0.3941 0.4277 0.4091 0.3582 0.3464 0.3388(20,20,20,20) 0.3779 0.7176 0.7554 0.7532 0.6853 0.6015 0.5864(5,10,10,20) 0.0651 0.3311 0.3435 0.3354 0.3230 0.2854 0.2750(20,10,10,5) 0.3212 0.5335 0.5511 0.5407 0.4283 0.4391 0.4382

Laplace Distribution(σ1, σ2, σ3, σ4) = (log 2, log 3, log 4, log 5)




(5,5,5,5) 0.0968 0.2668 0.2823 0.2815 0.2310 0.2738 0.2724(10,10,10,10) 0.2061 0.5037 0.5434 0.5358 0.4607 0.4687 0.4601(20,20,20,20) 0.5059 0.7911 0.8358 0.8342 0.7789 0.7484 0.7449(5,10,10,20) 0.0775 0.3853 0.4211 0.4138 0.3878 0.3527 0.3428(20,10,10,5) 0.4213 0.6039 0.6359 0.6289 0.4970 0.5337 0.5333

Normal Inverse Gaussian Distribution with Excess Kurtosis 7, Skewness 2(σ1, σ2, σ3, σ4) = (log 2, log 3, log 4, log 5)




(5,5,5,5) 0.0916 0.2391 0.2580 0.2546 0.2096 0.2594 0.2566(10,10,10,10) 0.1871 0.4583 0.4953 0.4838 0.4140 0.4280 0.4192(20,20,20,20) 0.4063 0.7218 0.7706 0.7703 0.7238 0.6162 0.6075(5,10,10,20) 0.0645 0.3596 0.3713 0.3747 0.3595 0.3476 0.3355(20,10,10,5) 0.3539 0.5404 0.5585 0.5434 0.4100 0.4366 0.4266


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Normal Distribution(σ1, σ2, σ3, σ4, σ5, σ6) = (1, 1, 2, 2, 2, 2)




(5,5,5,5,5,5) 0.1755 0.3903 0.4420 0.4522 0.3321 0.4311 0.4477(10,10,10,10,10,10) 0.4296 0.7137 0.7827 0.8082 0.6901 0.8137 0.8262(20,20,20,20,20,20) 0.8992 0.9530 0.9811 0.9891 0.9637 0.9924 0.9951

(5,5,10,10,20,20) 0.1560 0.3482 0.4514 0.4561 0.4464 0.4746 0.4760(20,20,10,10,5,5) 0.7781 0.9006 0.9142 0.9299 0.7705 0.8968 0.9122

Exponential Distribution(σ1, σ2, σ3, σ4, σ5, σ6) = (1, 1, 2, 2, 2, 2)




(5,5,5,5,5,5) 0.0997 0.2431 0.2822 0.2789 0.1983 0.2443 0.2425(10,10,10,10,10,10) 0.1897 0.4034 0.4707 0.4804 0.3515 0.4197 0.4358(20,20,20,20,20,20) 0.4583 0.7089 0.7597 0.7804 0.6891 0.6446 0.6556

(5,5,10,10,20,20) 0.0624 0.2094 0.2702 0.2737 0.2530 0.2702 0.2710(20,20,10,10,5,5) 0.4767 0.6235 0.6294 0.6543 0.4314 0.4814 0.4764

Laplace Distribution(σ1, σ2, σ3, σ4, σ5, σ6) = (1, 1, 2, 2, 2, 2)




(5,5,5,5,5,5) 0.1028 0.2993 0.3108 0.3202 0.2241 0.2938 0.2931(10,10,10,10,10,10) 0.2582 0.5073 0.5713 0.5887 0.4529 0.5257 0.5359(20,20,20,20,20,20) 0.6160 0.8117 0.8649 0.8905 0.8110 0.8133 0.8241

(5,5,10,10,20,20) 0.0637 0.2518 0.3182 0.3256 0.3173 0.3402 0.3462(20,20,10,10,5,5) 0.5896 0.7185 0.7350 0.7584 0.5173 0.6202 0.6277

Normal Inverse Gaussian Distribution with Excess Kurtosis 7, Skewness 2(σ1, σ2, σ3, σ4, σ5, σ6) = (1, 1, 2, 2, 2, 2)




(5,5,5,5,5,5) 0.1033 0.2457 0.2932 0.2928 0.1967 0.2764 0.2758(10,10,10,10,10,10) 0.2063 0.4486 0.5130 0.5232 0.3967 0.4586 0.4667(20,20,20,20,20,20) 0.5075 0.7448 0.8034 0.8154 0.7183 0.6698 0.6744

(5,5,10,10,20,20) 0.0618 0.2351 0.3070 0.3090 0.2870 0.3131 0.3235(20,20,10,10,5,5) 0.4769 0.6375 0.6573 0.6755 0.4367 0.4787 0.4749


Normal Distribution(σ1, σ2, σ3, σ4, σ5, σ6) = (log 2, log 3, log 4, log 5, log 6, log 7)




(5,5,5,5,5,5) 0.1815 0.5582 0.6371 0.6310 0.4865 0.5819 0.5884(10,10,10,10,10,10) 0.4638 0.8739 0.9444 0.9437 0.8780 0.9375 0.9350(20,20,20,20,20,20) 0.9272 0.9940 0.9995 0.9995 0.9981 0.9997 0.9995

(5,5,10,10,20,20) 0.2282 0.7150 0.8137 0.8045 0.7109 0.7236 0.7087(20,20,10,10,5,5) 0.7667 0.9417 0.9740 0.9731 0.9002 0.9686 0.9702

Exponential Distribution(σ1, σ2, σ3, σ4, σ5, σ6) = (log 2, log 3, log 4, log 5, log 6, log 7)




(5,5,5,5,5,5) 0.1052 0.3418 0.3945 0.3855 0.2874 0.3370 0.3282(10,10,10,10,10,10) 0.2045 0.5812 0.6596 0.6459 0.5350 0.5534 0.5504(20,20,20,20,20,20) 0.5116 0.8776 0.9373 0.9302 0.8888 0.8016 0.7933

(5,5,10,10,20,20) 0.0707 0.4380 0.4850 0.4768 0.4300 0.4179 0.4015(20,20,10,10,5,5) 0.4496 0.7144 0.7633 0.7518 0.5787 0.6120 0.5923

Laplace Distribution(σ1, σ2, σ3, σ4, σ5, σ6) = (log 2, log 3, log 4, log 5, log 6, log 7)




(5,5,5,5,5,5) 0.1099 0.4215 0.4497 0.4485 0.3400 0.4086 0.3991(10,10,10,10,10,10) 0.2819 0.7033 0.7846 0.7790 0.6631 0.6877 0.6798(20,20,20,20,20,20) 0.6748 0.9459 0.9793 0.9780 0.9564 0.9266 0.9261

(5,5,10,10,20,20) 0.0716 0.5294 0.5923 0.5938 0.5290 0.5140 0.5092(20,20,10,10,5,5) 0.5682 0.8024 0.8578 0.8489 0.6855 0.7656 0.7514

Normal Inverse Gaussian Distribution with Excess Kurtosis 7, Skewness 2(σ1, σ2, σ3, σ4, σ5, σ6) = (log 2, log 3, log 4, log 5, log 6, log 7)




(5,5,5,5,5,5) 0.1118 0.3492 0.4082 0.4013 0.2873 0.3727 0.3671(10,10,10,10,10,10) 0.2294 0.6311 0.7163 0.7029 0.5873 0.6003 0.5974(20,20,20,20,20,20) 0.5504 0.9048 0.9498 0.9411 0.9066 0.8136 0.8051

(5,5,10,10,20,20) 0.0732 0.4704 0.5355 0.5232 0.4717 0.4714 0.4701(20,20,10,10,5,5) 0.4608 0.7272 0.7900 0.7798 0.5949 0.6141 0.5951


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combination of levene-type tests and a finite-intersection method for testing equality ... ·...

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