estimation of parameters for exponentiated-weibull family under type-ii censoring scheme

Computational Statistics & Data Analysis 48 (2005) 509–523www.elsevier.com/locate/csda

Estimation of parameters forexponentiated-Weibull family under type-II

censoring scheme

Umesh Singh, Pramod K. Gupta∗ , S.K. UpadhyayDepartment of Statistics, Banaras Hindu University, Varanasi 221 005, India

Received 22 January 2004; received in revised form 20 February 2004; accepted 21 February 2004

Abstract

Bayes and classical estimators have been obtained for two-parameter exponentiated-Weibulldistribution when sample is available from type-II censoring scheme. Bayes estimators havebeen developed under squared error loss function as well as under LINEX loss function usingnon-informative type of priors for the parameters. Besides, the generalized maximum likelihoodestimators and the usual maximum likelihood estimators have also been attempted. It has beenseen that the estimators obtained are not available in nice closed forms, although they can beeasily evaluated for the given sample by using suitable numerical methods. The performance ofthe proposed estimators have been compared on the basis of their simulated risks (average lossover the sample space) obtained under squared error as well as under LINEX loss functions.c© 2004 Elsevier B.V. All rights reserved.

Keywords: Bayes estimators; Generalized maximum likelihood estimator; Maximum likelihood estimator;Non-informative type priors; Type-II censoring; Squared error loss function; LINEX loss function; Risk

1. Introduction

Exponentiated-Weibull distribution (EWD) was @rst introduced by Mudholkarand Hutson (1996) as a simple generalization of the well-known Weibull familyby introducing one more shape parameter. The probability density function and the

∗ Corresponding author. Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, ROC.Tel.: +886-227-835-611-209; fax: +886-227-831-523.

E-mail address: pk [email protected] (P.K. Gupta).

0167-9473/$ - see front matter c© 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.csda.2004.02.009

mailto:[email protected]

510 U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509–523

distribution function of EWD are expressed as

f(y) = ��(1 − e−y�)�−1e−y�y�−1; �; �¿ 0; 0¡y¡∞ (1)

and

F(y) = (1 − e−y�)�; (2)

respectively, where � and � are the shape parameters of the model. The distinguishedfeature of EWD from other lifetime distribution is that it accommodates nearly alltypes of failure rates both monotone and non-monotone (unimodal and bathtub) andincludes a number of distributions as particular cases. The structural properties ofEWD have been discussed by Mudholkar and Hutson (1996). It is worthwhile tomention here that the lifetime data following constant and monotone (increasing anddecreasing) type of failure rates are well described by the exponential and the Weibulldistributions, respectively. It may be further noted that the inferential procedures basedon these models are often simple and exist in closed forms. But due to expeditiousimprovement in the techniques of science and technology, the non-monotone failurerate is increasingly becoming common in the @eld of engineering, medical and spaceexploration and, therefore, the aforesaid models are no longer justi@ed for their use.However, a number of other lifetime distributions are also available in the literatureswhich any how serve the need when lifetime data show non-monotone failure rates.Generalized Weibull, generalized Rayleigh, generalized gamma, generalized F, mixtureof Weibull distributions, lognormal and loglogistic, etc. (see Mudholkar and Hutson,1996 for details) are few examples. But, the inferential procedures for these exempli@edmodels, as studied and discussed by Bain (1974); Gore et al. (1986) and Lawless(1982), often present diKculties, especially in the presence of censoring. In contrast tothese distributions, EWD enjoys the advantage of being parsimonious in parameter andhence estimation of parameters of this model is expected not to pose much mathematicalcomplexity, see, Singh et al. (2002) and may work well in the case of censoring asanticipated by Mudholkar and Hutson (1996). The estimation procedure for EWD undercensoring case seems to be untouched and, therefore, we are interested to develop theestimation procedure for EWD for censored sample case. For simplicity, we shall,however, be con@ned to type-II censored data only (see Lawless, 1982).

On another important issue, it is to be noted that the inferential procedures forlifetime models are often developed using squared error loss function (SELF). Nodoubt, the use of SELF is well justi@ed when the loss is symmetric in nature. Its useis also very popular, perhaps, because of its mathematical simplicity. But in life testingand reliability problems, the nature of losses are not always symmetric and hence theuse of SELF is forbidden and unacceptable in many situations. Inappropriateness ofSELF has also been pointed out by diLerent authors. Ferguson (1967), Zellner andGeisel (1968) Aitchison and Dunsmore (1975), Varian (1975) and Berger (1980) arefew among many others. It is because of this fact that Varian (1975) introduced LINEXloss function (LLF) which is the simple generalization of SELF and can be used inalmost every situation. SELF can also be considered as particular case of LLF (seeZellner, 1986; Parsian, 1990; Khatree, 1992, etc.). LLF is de@ned as

L() = b(ea − a− 1); a �= 0; b¿ 0; (3)

U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509–523 511

where ‘a’ and ‘b’ are the shape and scale parameters of the loss function (3). Obviously,the nature of LLF changes according to the choice of a.

The aim of this study is to develop point estimators for both the shape parametersof the EWD. In the next section, @rst of all the usual maximum likelihood estima-tors are discussed. Then the Bayes estimators under SELF as well as under the LLFare obtained. Lastly, the generalized maximum likelihood estimators for the parametersunder the same Bayesian setup have also been discussed. Section 3 contains an illus-trative examples based on both real and simulated data sets. The estimators obtainedin Section 2 are not reducible in nice closed forms but they can be easily evaluatedusing suitable numerical methods. The performances of the estimators are, therefore,compared on the basis of their simulated risks obtained under both SELF and LLFseparately and are summarized in the last but one section. The last section contains abrief conclusion on the use of the estimators.

2. Estimation of parameters

Suppose that n items, whose life times follow EWD, are put on test. Due to the costand time considerations, the test is terminated as soon as the rth (r6 n) item fails.The lifetimes of these @rst r failed items say y = (y1; y2; : : : ; yr) are observed. Thelikelihood function corresponding to this set-up can, therefore, be easily written as

l(y=�; �) =n!

(n− r)! (��)rr∏i=1

y�−1i

r∏i=1

e−y�ir∏i=1

(1 − e−y�i )�−1

×[1 − (1 − e−y�r )�

]n−r: (4)

2.1. Maximum likelihood estimator (MLE)

The logarithm of the likelihood function given in Eq. (4) can be expressed as

L(y=�; �) = logn!

(n− r)! + rlog (��) + (�− 1)r∑i=1

log(yi) −r∑i=1

y�i + (�− 1)

×r∑i=1

log(1 − e−y�i ) + (n− r)log[1 − (1 − e−y�r )�

]: (5)

To obtain the normal equations for the unknown parameters, we diLerentiate (5) par-tially with respect to the parameters � and � and equate to zero. The resulting equationsare given below in (6) and (7), respectively,

0 =@L@�

=r�

+r∑i=1

log(yi) −r∑i=1

y�i log(yi) + (�− 1)r∑i=1

e−y�i y�i log(yi)(1 − e−y�i )


− (n− r)�(1 − e−y�r )�−1e−y�r y�r log(yr)[1 − (1 − e−y�r )�

] ; (6)

0 =@L@�

=r�

+r∑i=1

log(1 − e−y�i

)− (n− r)(1 − e−y�r )� log(1 − e−y�r )[

1 − (1 − e−y�r )�] : (7)

The solutions of above equations are the MLEs of the EWD parameters � and � denotedas Q�ml and Q�ml, respectively. It may be noted here that the equations expressed in (6)and (7) cannot be solved analytically and, therefore, we suggest to use iterative methodsfor @nding the numerical solutions of these equations. The solutions of these equationshave been obtained using C05PCF routine of Nag (1993) which uses Powell hybridtype Newton Raphson method and provides the global solution. The routine requiresall second-order derivatives with respect to � and � which can be easily obtained from(6) and (7).

2.2. Bayes estimators

Consider independent non-informative (or vague) type of priors for the parameters� and � as

g1(�) =1c; 0¡�¡c; (8)

g2(�) =1�; �¿ 0: (9)

Combining (8) and (9) with Eq. (4) and using Bayes theorem, the joint posteriordistribution is derived as follows:

∏(�; �|y) =

1j1�r� r−1

r∏i=1

e−y�ir∏i=1

y�−1i

r∏i=1

(1 − e−y�i )�−1

×[1 − (1 − e−y�r )�

]n−r; (10)

where

j1 =∫ c

0

∫ ∞

0�r� r−1

r∏i=1

e−y�ir∏i=1

y�−1i

r∏i=1

(1 − e−y�i )�−1

×[1 − (1 − e−y�r )�

]n−rd� d�: (11)

Marginal posterior of a parameter is obtained by integrating the joint posterior distri-bution with respect to the other parameter and hence the marginal posterior of � canbe written, after simpli@cation, as

∏(�|y) =

�r

j1

r∏i=1

e−y�ir∏i=1

y�−1i j2; (12)


where

j2 =∫ ∞

0� r−1

r∏i=1

(1 − e−y�i )�−1[1 − (1 − e−y�r )�

]n−rd�: (13)

Similarly integrating the joint posterior with respect to �, the marginal posterior � canbe obtained as

∏(�|y) =

� r−1j3j1

; (14)

where

j3 =∫ c

0�r

r∏i=1

e−y�ir∏i=1

y�−1i

r∏i=1

(1 − e−y�i )�−1[1 − (1 − e−y�r )�

]n−rd�: (15)

2.3. Bayes estimator under squared error loss function (BESF)

The Bayes estimators for parameters � and � of EWD may be de@ned as

Q�bs = E(�=y) =∫ c

0�

∏(�=y) d�;

Q�bs = E(�=y) =∫ ∞

0�

∏(�=y) d�;

respectively. These estimators can be expressed as

Q�bs =j4j1

(16)

and

Q�bs =j5j1; (17)

where

j4 =1j1

∫ c

0

∫ ∞

0�r+1� r−1

r∏i=1

e−y�ir∏i=1

y�−1i

r∏i=1

(1 − e−y�i )�−1

×[1 − (1 − e−y�r )�

]n−rd� d� (18)

and

j5 =1j1

∫ c

0

∫ ∞

0(��)r

r∏i=1

e−y�ir∏i=1

y�−1i

r∏i=1

(1 − e−y�i )�−1

×[1 − (1 − e−y�r )�

]n−rd� d�: (19)

It may be noted here that the BESFs are not reducible in nice closed forms; however,we propose to use 16-point Gaussian quadrature formulas for their evaluation.


2.4. Bayes estimator under LINEX loss function (BELF)

Following Zellner (1986), the Bayes estimators for the shape parameters � and � ofEWD under LLF are

Q�bl = −1alog (E(e−a�))

and

Q�bl = −1a

log(E(e−a�));

respectively, where E(·) denotes the posterior expectation. After simpli@cation, we have

Q�bl = −1a

log(j6j1

)(20)

and

Q�bl = −1a

log(j7j1

); (21)

where j1 is given in (11),

j6 =1j1

∫ c

0

∫ ∞

0�r� r−1e−a�

r∏i=1

e−y�ir∏i=1

y�−1i

r∏i=1

(1 − e−y�i )�−1

×[1 − (1 − e−y�r )�

]n−rd� d� (22)

and

j7 =1j1

∫ c

0

∫ ∞

0�r� r−1e−a�

r∏i=1

e−y�ir∏i=1

y�−1i

r∏i=1

(1 − e−y�i )�−1

×[1 − (1 − e−y�r )�

]n−rd� d�: (23)

As mentioned earlier, the integrals involved in (21) and (22) are not solvable analyt-ically and, therefore, the solution can be obtained using 16-point Gaussian quadratureformulas.

2.5. Generalized maximum likelihood estimator (GMLE)

The GMLE of a parameter is the value of the parameter that maximizes the concernedmarginal posterior density (see Martz and Waller, 1982) for details. The marginal pos-terior densities of the parameters � and � are expressed by (12) and (14), respectively,which are not available in closed forms and hence the exact analytical expressions forGMLEs do not exist. However, GMLEs of the parameters can be obtained with thehelp of numerical iterative methods without much diKculty. The results reported hereare based on the procedure developed by Singh et al. (2002) which provides globalmaxima for the concerned posteriors. The estimators of the parameters of � and � aredenoted by Q�GML and Q�GML, respectively.


3. Numerical illustration

Example 3.1. To illustrate the usefulness of the proposed estimators obtained inSection 2 with real situations, we considered here the real data-set initially reported byAarset (1987) to identify the bathtub hazard rate contains lifetime of 50 devices.Mudholkar and Srivastava (1993) used this in context of three-parameter EWD tostudy the suitability of the model with bathtub hazard rate. Hence we obtained theproposed estimators for Aarset data and summarized it in Table 1.

Bayes estimators have evaluated for the prior hyper-parameter c= 4, 10 and 12 andtheir corresponding values have shown in Table 1. Table 1 revealed that the Bayesestimators are not seems very sensitive with variation of ‘c’. It is also worth mentionedthat though the Bayes estimators developed with non-informative prior (vague) yet theestimated values of Bayes estimators are not very far from the estimated values ofMLE. Obviously, we do not expect much to conclude from this reanalysis, perhaps weare capable to show that the proposed estimators can be easily obtained in practicalsituations in spite of non-existence of their closed form solutions.

Example 3.2. Next, we generated a sample of size 10 from the EWD with parameters� = 2:0 and � = 0:5. The considered values of � and � are meant for illustration onlyand other values can also be taken for generating the samples from EWD. In order toget a type II censored data, the generated observations were ordered and the largestfour observations were removed so that the observed failures consist of @rst 6 itemsonly. The observed life times obtained in this way are reported below:

0:0673 0:1293 0:1878 0:1879 0:2454 0:4117

Table 2 given shows the diLerent estimators for a= 1:0, 0.01 and −1:0, c= 4, 10, 12;n=10 and r=6 (r=n=0:6). It may be seen from Table 2 that Bayes estimators, (BESFand BELF with a= 1:0) are close to the true values of � and � as compared to MLEand GMLE. The change in the values of ‘a’ does eLect the BELF estimates only. Buton the basis of a single sample estimate, it will be illogical and inappropriate to inferthat BESF and BELF perform better than MLE and GMLE. One way to study theperformances of these estimators would be to study their behavior for long term use.

Table 1Estimates of � and � (c = 4; 10; 12; n = 50) for Aarset (1987) data

Hyper-parameter c = 4 c = 10 c = 12

Estimators � � � � � �

GMLE 0.290 6.550 0.290 6.730 0.290 6.790BESF 0.268 6.665 0.271 6.745 0.312 6.798BELF (a= 1:0E − 05) 0.268 6.665 0.271 6.745 0.313 6.798BELF (a= 1:0) 0.268 6.256 0.277 6.419 0.322 6.717BELF (a= −1:0) 0.268 7.153 0.277 7.366 0.322 7.649

MLE 0.276 6.826


Table 2Estimates of � and � for c = 4, 10, 12, n = 10 and r=n = 0:6 for Simulated data

Hyper-parameter c = 4 c = 10 c = 12

Estimators � � � � � �

GMLE 2.752 0.119 2.767 0.176 2.767 0.176BESF 2.460 0.406 2.522 0.418 2.528 0.421BELF (a= 1:0E − 03) 2.460 0.406 2.522 0.418 2.528 0.421BELF (a= −1:0) 2.581 0.450 2.614 0.459 2.619 0.461BELF (a= 1:0) 2.034 0.378 2.107 0.401 2.108 0.400

MLE 7.494 0.112

In other words, we propose to study the behavior of these estimators on the basis oftheir risks (expected loss over whole sample space) which is given in the next section.

4. Comparison

The estimators, developed in Section 2, are studied here on the basis of their risksobtained under two diLerent loss functions, namely, SELF and LLF. Risks of theestimators have been estimated on the basis of 5000 randomly generated samples ofsize 10 for various combinations of diLerent parameters. It is to be noted that both �and � were given arbitrary choice for the generation of random samples from the EWDalthough in a Bayesian framework the assumed prior distributions should normally beused for generating the concerned parameters. In our situation since the prior for �is improper, it cannot be used for generation of �. For �, however, one can consideruniform prior in the range (0; c) but in order to maintain the uniformity in both � and�, the parameter � was also given arbitrary choice.

Thus, we considered both (�; �) = (0:5 (0:5) 2:5). These values were chosen soas to accommodate all types of failure rates like monotone, and non-monotone andalso perhaps to cover the situations where complexities are expected to occur in thecalculation of MLEs (see, for example, Cheng and Amin, 1983). The censoring frac-tion r=n, hyper-parameter c (involved only in the risks of Bayes estimators) and theshape parameter ‘a’ of the LLF were considered, respectively, as r=n= (0:4 (0:2) 1:0),c = (4 (2) 12) and a = −1:0, 0.01, 1.0. Further, while studying the eLect of thehyper-parameter c on the risks of Bayes estimators and GMLE, it was noticed thatvariation in the values of c has negligible eLect on the trend of risk and, therefore,@gures have been shown for c = 4 only.

Fig. 1 (S-1–S-8) and Fig. 2 (L-1–L-16) summarize the results partially althoughmost of the reported @ndings are based on all detailed evaluations. It may be fur-ther noted that the scale on y-axis varies from @gure to @gure. The @ndings are pre-sented below, under two diLerent situations. The @rst one is the case when over- andunder-estimation are considered to be of an equal importance, that is, the situation


S-1 r/n=0.4; θ=0.5

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α

Ris

k

MLE GMLE BESF BELF(-1)BELF(1) BELF(0)

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α


S-2 r/n=0.4; θ=2.0

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α


5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α


Ris

k

Ris

k

Ris

k

S-3 r/n=0.8; θ =0.5

1.00E-01

1.00E+00

1.00E+01

1.00E+02S-4 r/n=0.8; θ =2.0

1.00E-01

1.00E+00

1.00E+01

1.00E+02

Risk of Estimators of � under SELF

Risk of Estimators of θ under SELF

S-5 r/n=0.4; α=0.5

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Ris

kR

isk

Ris

k

MLE GMLE BESF BELF(-1)

BELF(1) BELF(0)

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ


BELF(1) BELF(0)

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ


BELF(1) BELF(0)

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ


BELF(1) BELF(0)

S-6 r/n=0.4; α =2.0

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

S-7 r/n=0.8; α =0.5 S-8 r/n=0.8; α=2.0

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

Ris

k

Fig. 1. (S-1–S-8) Risk of estimators under SELF. (L-1–L-8) Risk of estimators of � under LLF when a=1:0.


Risk of Estimators of �� under LLF when a = 1.0

L-1 r/n=0.4; θ=0.5

1.00E-021.00E+011.00E+041.00E+071.00E+101.00E+131.00E+161.00E+191.00E+221.00E+251.00E+281.00E+311.00E+34

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α

Ris

k

MLE GMLE BESF BELF

L-2 r/n=0.4; θ=2.0

1.00E-021.00E+031.00E+081.00E+131.00E+181.00E+231.00E+281.00E+331.00E+381.00E+431.00E+481.00E+531.00E+581.00E+63

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α

Ris

k

MLE GMLE BESF BELF

L-3 r/n=0.8; θ=0.5

1.00E-02

1.00E+00

1.00E+02

1.00E+04

1.00E+06

1.00E+08

1.00E+10

1.00E+12

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α

Ris

k

MLE GMLE BESF BELF

L-4 r/n=0.8; θ=2.0

1.00E-02

1.00E+021.00E+06

1.00E+10

1.00E+14

1.00E+181.00E+22

1.00E+26

1.00E+30

1.00E+34

1.00E+38

1.00E+42

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α

Ris

k

MLE GMLE BESF BELF

Risk of Estimators of � under LLF when a = -1.0

L-5 r/n=0.4; θ-0.5

1.00E-02

1.00E-01

1.00E+00

1.00E+01

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α

Ris

k

MLE GMLE BESF BELF

L-6 r/n=0.4; θ=2.0

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α

Ris

k

MLE GMLE BESF BELF

L-7 r/n=0.8; θ=0.5

1.00E-02

1.00E-01

1.00E+00

1.00E+01

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α

Ris

k

MLE GMLE BESF BELF

L-8 r/n=0.8; θ=2.0

1.00E-02

1.00E-01

1.00E+00

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

α

Ris

k

MLE GMLE BESF BELF

Fig. 2. (L-1–L-16) Risk of estimators under LLF.


Risk of Estimators of � under LLF when a = 1.0

L-9 r/n=0.4; α=0.5

1.00E-02

1.00E+01

1.00E+04

1.00E+07

1.00E+10

1.00E+13

1.00E+16

1.00E+19

1.00E+22

1.00E+25

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Ris

k

MLE GMLE BESF BELF

L-10 r/n=0.4; α=2.0

0.01

10

10000

1E+07

1E+10

1E+13

1E+16

1E+19

1E+22

1E+25

1E+28

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Risk

MLE GMLE BESF BELF

L-11 r/n=0.8; α=0.5

1.00E-02

1.00E+00

1.00E+02

1.00E+04

1.00E+06

1.00E+08

1.00E+10

1.00E+12

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Risk

MLE GMLE BESF BELF

L-12 r/n=0.8; α=2.0

1.00E-02

1.00E+00

1.00E+02

1.00E+04

1.00E+06

1.00E+08

1.00E+10

1.00E+12

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Risk

MLE GMLE BESF BELF

Risk of Estimators of � under LLF when a = 1.0

L-13 r/n=0.4; α=0.5

1.00E-02

1.00E-01

1.00E+00

1.00E+01

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Ris

k

MLE GMLE BESF BELF

L-14 r/n=0.4; α=2.0

1.00E-02

1.00E-01

1.00E+00

1.00E+01

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Ris

k

MLE GMLE BESF BELF

L-15 r/n=0.8; α=0.5

1.00E-02

1.00E-01

1.00E+00

1.00E+01

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Ris

k

MLE GMLE BESF BELF

L-16 r/n=0.8; α=2.0

1.00E-02

1.00E-01

1.00E+00

1.00E+01

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Ris

k

MLE GMLE BESF BELF

Fig. 2. continued.


L-17 r/n=0.8; α=2.0; a=1.0

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Ris

k

MLE BESF BELF GMLE

L-18 r/n=0.8; θ=2.0; a = -1.0

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Ris

k

MLE BESF BELF GMLE

Risk of the Various Estimators of � Under LLF for a = 1.0 and a= -1.0

L-19 r/n=0.8; α=2.0; a=1.0

1.00E-03

1.00E-02

1.00E-01

1.00E+00

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Ris

k

MLE BESF BELF GMLE

L-20 r/n=0.8; α=2.0; a = -1.0

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00

θ

Ris

k

MLE BESF BELF GMLE

Risk of the VariousEstimators of � Under LLF for a = 1.0 and a = -1.0

Fig. 2. continued.

where use of symmetric loss function is justi@ed. The second one concerns with thesituation where over- and under-estimation are of an unequal importance, that is, whenthe asymmetric nature of loss is justi@ed.

4.1. Over-estimation and under-estimation are of equal importance

As mentioned earlier when over- and under-estimation are of equal importance, useof symmetric loss is most justi@ed and, therefore, for comparing the performance andstudying the eLect of various constants, we are considering below the risks under SELF(see Fig. 1 (S-1–S-8)). Here risks of the estimators are obtained under SELF and henceone can agree that BESF is more justi@ed estimators than BELF. It may be interestingto see whether BESF outperforms BELF for SELF.E<ect of r/n: As r=n increases, the risk of all the estimators decreases. However,

in general, the rate of decrement in the risks are more for the estimators of � thanthose of �. Risks of the estimators obtained by diLerent methods also show diLerentrate of decrements. Among the various estimators of �, the rate of decrement forBELF6BESF¡MLE¡GMLE whereas for the estimators of � the trend in rate ofchange of concerned risks are similar to those of � except for MLE and GMLE. Therate of increase in the risk of GMLE is less than that of MLE (see Fig. 1 S-1 andS-3, S-2 and S-4, S-5 and S-7, S-6 and S-8).E<ect of �: As � increases risks of MLE and GMLE increase while risk of the

BESF and BELF decrease in general but when r=n and � become large, the risks ofBESF and BELF early increase and then decrease (see Fig. 1 (S-1–S-4)). It has alsobeen noted that increase in � results in a slight decrease in the magnitude of the risks


of the estimators of � but the relative performance remains the same (see Fig. 1 S-1and S-2, S-3 and S-4).E<ect of �: Risk of almost all the estimators of � increases as the value of parameter

increases. The increment in the magnitude of the risk is found to be high for MLEand BELF with a= −1:0 while increment in other estimators are small in comparisonto MLE and BELF with a= −1:0 (see Fig. 1(S-5–S-8)). Risks of the estimators � donot vary much with variation in � (see two consecutive combinations of Fig. 1 fromS-5 to S-8).Comparison of various estimators: Bayes estimators for BESF and BELF for �

always provide smaller risks than those of MLE and GMLE. It is interesting to notehere that BELF with a=1:0 have smallest risk among all, though risk of BESF is alsoclose to it. The risk of GMLE is largest over the considered parameter space whilerisk of MLE is less than that of GMLE but close to those of GMLE. The diLerencebetween risk of MLE (or GMLE) and risk of Bayes estimators increases as � increases.It is also noted that among Bayes estimators the risk of BELF with a= −1:0 is foundto be largest (see Fig. 1 (S-1–S-4)).

The risk of BELF with a= 1:0 has smallest risk among all estimators of parameter� while risk of BELF with a= −1:0 has largest risk among all others. Risks of otherestimators are more or less close to the risk of BELF with a = 1:0 and on the basisof the magnitude of risk, the relative position of various estimators can be expressedas BESF¡GMLE¡MLE. It is also noted that the diLerence between risk of theestimators of � decreases as r=n increases (see Fig. 1 (S-5–S-8)).

4.2. Over-estimation and under-estimation are not of equal importance

Asymmetric loss function is most justi@ed loss function to deal with such situa-tions and, therefore, LLF will be considered here due to its properties as discussed inSection 1. It is to be mentioned again here that the nature of LLF changes accord-ing to its shape parameter a. Since ‘a’ is involved in the expression of BELF, theestimators of � and � obtained under BELF also change with changes in a. We, there-fore, consider only the appropriate BELF estimator that matches with the de@nitionof LLF. That is, when over-estimation/under-estimation becomes more serious thanunder-estimation/over-estimation, the risk of the estimators obtained under LLF withpositive/negative choice of shape parameter a is justi@ed and according to Zellner(1986), in this case, the most appropriate estimator is Bayes estimator under LLF witha¿ 0=a¡ 0. Since LLF becomes quite asymmetric with a=1:0 and −1:0 (see Zellner,1986) we considered, therefore, a= 1:0 and −1:0 only.E<ect of r/n: As usual, the risk of the estimators of the parameters decreases as

r=n increases although the eLect of r=n on estimators is more or less same as notedfor risks obtained under SELF (see even and odd combinations of Fig. 2 from L-1 toL-16).E<ect of �: Generally as � increases, risks of the Bayes estimators do not change

much for changes in the values of �, however, the risk of MLE and GMLE increaseswhen over-estimation is more serious than under-estimation (see Fig. 2 (L-1–L-4)).When under-estimation becomes more serious than over-estimation, i.e., when a=−1:0,


risk of Bayes estimators decreases in general as � increases while the risk of MLE andGMLE increases (see Fig. 2 (L-6–L-8)).E<ect of �: The trend of the risks of the estimators of � is nearly same as the

trend of the risks of the estimators of � for the cases when over-estimation is moreserious (a=1:0); see Fig. 2 (L-9–L-12). But the circumstances where under-estimationbecomes more serious (a=−1:0) the risk of all the estimators increases as � increaseswhich can be seen from Fig. 2 L-13 to L16.Comparison of various estimators: When a = 1:0; this is the case where over-

estimation is more serious than under-estimation. BELF of the parameters show smallestrisk although the risk of BESF is near to the risk of BELF. The diLerence betweenthe risks of BESF and BELF is very small for � in comparison that of � (see Fig. 2(L-1–L-4) and L-9–L-12). Risk of GMLE for the parameter � is to be found morethan that of corresponding MLE (see Fig. 2 (L-1–L-4)) while the risk of GMLE for� shows smaller risk than that of MLE (see Fig. 2 (L-9–L-12)).

When (a = −1:0) under-estimation is considered to be more serious than over-estimation, the risk of the BESF is smallest among the risk of all other estimatorswhile the risk of BELF is very close to it (see Fig. 2 (L-5–L-8) and (L-13–L-16)).The trend of the risk of the GMLE and MLE is almost same as discussed earlier fora= 1:0.

Finally, to complete the study, we considered to evaluate the risks of various esti-mators of � and � under LLF for a = 1:0 and −1:0 when the sample size n is large(n = 100). The results are shown for the purpose of illustration only although it isoften seen that classical MLEs may outperform the Bayes estimators for large valuesof n. Fig. 2 (L-17–L-20) show the risk of various estimators of � and � under LLFfor a=1:0 and −1:0. It is obvious that the risk of all the estimators of � is very closeto each other (see Fig. 2 L-17 and L-18) while for � the risks of MLE and GMLEare smaller as compared to those of BESF and BELF (see Fig. 2 L-19 and L-20). Sofor �, Bayes estimators can still be used whereas for �, one should preferably considerthe MLE or GMLE.

5. Conclusions

The comprehensive comparison of the risks of the estimators and eLects of concernedparameters on their risks disclose that if the samples are highly censored, the BESFand BELF give smaller risks than those of MLE and GMLE. However, the risks ofGMLE and MLE come closer to the risks of BELF and BESF for small censoringfractions. GMLE for � always has greater risk in comparison to the MLE for all theconsidered values while the risk of GMLE for � has smaller value than that of MLE.GMLE comes very near to the Bayes estimators when under-estimation is consideredmore serious than over-estimation. The BELF of � with a= −1:0 under SELF showshigher risk among all other estimators of �. Therefore, for the estimation of parameter�, the use of BELF with a = 1:0 may be proposed. BESF can also be used for theestimation of � as the risk of BESF and BELF are not much diLerent. However, forestimation of � one can use safely either GMLE or BESF.


Acknowledgements

Authors are thankful to the co-editor and referees for their comments and suggestionsthat helped to improve the earlier version of the paper. The research work of Pramod K.Gupta was @nancially supported by the Council of Scienti@c and Industrial Research,Government of India, Senior Research Fellowship Grant.

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estimation of parameters for exponentiated-weibull family under type-ii censoring scheme

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