Estimation of parameters for exponentiated-Weibull family under type ...
Computational Statistics & Data Analysis 48 (2005) 509 – 523 www.elsevier.com/locate/csda
Estimation of parameters for exponentiated-Weibull family under type-II censoring scheme Umesh Singh, Pramod K. Gupta∗ , S.K. Upadhyay Department 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-Weibull distribution when sample is available from type-II censoring scheme. Bayes estimators have been developed under squared error loss function as well as under LINEX loss function using non-informative type of priors for the parameters. Besides, the generalized maximum likelihood estimators and the usual maximum likelihood estimators have also been attempted. It has been seen that the estimators obtained are not available in nice closed forms, although they can be easily evaluated for the given sample by using suitable numerical methods. The performance of the proposed estimators have been compared on the basis of their simulated risks (average loss over 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 Mudholkar and Hutson (1996) as a simple generalization of the well-known Weibull family by 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). c 2004 Elsevier B.V. All rights reserved. 0167-9473/$ - see front matter doi:10.1016/j.csda.2004.02.009
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distribution function of EWD are expressed as
f(y) = (1 − e−y )−1 e−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 distinguished feature of EWD from other lifetime distribution is that it accommodates nearly all types of failure rates both monotone and non-monotone (unimodal and bathtub) and includes a number of distributions as particular cases. The structural properties of EWD have been discussed by Mudholkar and Hutson (1996). It is worthwhile to mention here that the lifetime data following constant and monotone (increasing and decreasing) type of failure rates are well described by the exponential and the Weibull distributions, respectively. It may be further noted that the inferential procedures based on these models are often simple and exist in closed forms. But due to expeditious improvement in the techniques of science and technology, the non-monotone failure rate is increasingly becoming common in the @eld of engineering, medical and space exploration 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 literatures which any how serve the need when lifetime data show non-monotone failure rates. Generalized Weibull, generalized Rayleigh, generalized gamma, generalized F, mixture of Weibull distributions, lognormal and loglogistic, etc. (see Mudholkar and Hutson, 1996 for details) are few examples. But, the inferential procedures for these exempli@ed models, 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 to these distributions, EWD enjoys the advantage of being parsimonious in parameter and hence estimation of parameters of this model is expected not to pose much mathematical complexity, see, Singh et al. (2002) and may work well in the case of censoring as anticipated by Mudholkar and Hutson (1996). The estimation procedure for EWD under censoring case seems to be untouched and, therefore, we are interested to develop the estimation 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 for lifetime models are often developed using squared error loss function (SELF). No doubt, the use of SELF is well justi@ed when the loss is symmetric in nature. Its use is also very popular, perhaps, because of its mathematical simplicity. But in life testing and reliability problems, the nature of losses are not always symmetric and hence the use of SELF is forbidden and unacceptable in many situations. Inappropriateness of SELF has also been pointed out by diLerent authors. Ferguson (1967), Zellner and Geisel (1968) Aitchison and Dunsmore (1975), Varian (1975) and Berger (1980) are few among many others. It is because of this fact that Varian (1975) introduced LINEX loss function (LLF) which is the simple generalization of SELF and can be used in almost every situation. SELF can also be considered as particular case of LLF (see Zellner, 1986; Parsian, 1990; Khatree, 1992, etc.). LLF is de@ned as L( ) = b(ea − a − 1);
a = 0;
b ¿ 0;
(3)
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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 parameters of the EWD. In the next section, @rst of all the usual maximum likelihood estimators are discussed. Then the Bayes estimators under SELF as well as under the LLF are obtained. Lastly, the generalized maximum likelihood estimators for the parameters under the same Bayesian setup have also been discussed. Section 3 contains an illustrative examples based on both real and simulated data sets. The estimators obtained in Section 2 are not reducible in nice closed forms but they can be easily evaluated using suitable numerical methods. The performances of the estimators are, therefore, compared on the basis of their simulated risks obtained under both SELF and LLF separately and are summarized in the last but one section. The last section contains a brief 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 cost and time considerations, the test is terminated as soon as the rth (r 6 n) item fails. The lifetimes of these @rst r failed items say y = (y1 ; y2 ; : : : ; yr ) are observed. The likelihood function corresponding to this set-up can, therefore, be easily written as r
l(y=; ) =
r
r
i=1
i=1
n! ()r yi−1 e−yi (1 − e−yi )−1 (n − r)! i=1
× 1 − (1 − e−yr )
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=; ) = log
r
r
i=1
i=1
n! log(yi ) − yi + ( − 1) + rlog () + ( − 1) (n − r)!
r
×
log(1 − e−yi ) + (n − r)log 1 − (1 − e−yr ) :
(5)
i=1
To obtain the normal equations for the unknown parameters, we diLerentiate (5) partially with respect to the parameters and and equate to zero. The resulting equations are given below in (6) and (7), respectively, 0=
r
r
r
i=1
i=1
i=1
e−yi y log(yi ) @L r i = + log(yi ) − yi log(yi ) + ( − 1) @ (1 − e−yi )
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(n − r)(1 − e−yr )−1 e−yr yr log(yr ) ; 1 − (1 − e−yr ) r (n − r)(1 − e−yr ) log(1 − e−yr ) @L r = + : log 1 − e−yi − 0= @ 1 − (1 − e−yr ) i=1 −
(6) (7)
The solutions of above equations are the MLEs of the EWD parameters and denoted as Qml and Qml , 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 methods for @nding the numerical solutions of these equations. The solutions of these equations have been obtained using C05PCF routine of Nag (1993) which uses Powell hybrid type Newton Raphson method and provides the global solution. The routine requires all 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 1 g1 () = ; c
0 ¡ ¡ c;
(8)
1 g2 () = ;
¿ 0:
(9)
Combining (8) and (9) with Eq. (4) and using Bayes theorem, the joint posterior distribution is derived as follows: r r r 1 yi−1 (1 − e−yi )−1 (; |y) = r r−1 e−yi j1 i=1
i=1
× 1 − (1 − e−yr ) where
j1 =
c
0
0
∞
r r−1
r i=1
× 1 − (1 − e−yr )
e−yi
n−r
r
n−r
yi−1
i=1
d d:
i=1
;
(10)
r (1 − e−yi )−1 i=1
(11)
Marginal posterior of a parameter is obtained by integrating the joint posterior distribution with respect to the other parameter and hence the marginal posterior of can be written, after simpli@cation, as r r r −yi −1 (|y) = e yi j2 ; j1 i=1
i=1
(12)
U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523
where
j2 =
∞
0
r−1
r n−r (1 − e−yi )−1 1 − (1 − e−yr ) d:
513
(13)
i=1
Similarly integrating the joint posterior with respect to , the marginal posterior can be obtained as r−1 j3 (|y) = ; (14) j1 where
j3 =
c
r
0
r r r n−r e−yi yi−1 (1 − e−yi )−1 1 − (1 − e−yr ) d: i=1
i=1
(15)
i=1
2.3. Bayes estimator under squared error loss function (BESF) The Bayes estimators for parameters and of EWD may be de@ned as c Qbs = E(=y) = (=y) d; 0 ∞ Qbs = E(=y) = (=y) d; 0
respectively. These estimators can be expressed as j4 Qbs = j1
(16)
and j5 Qbs = ; j1 where j4 =
1 j1
(17)
c
0
∞
0
r+1 r−1
i=1
× 1 − (1 − e−yr ) and j5 =
1 j1
0
c
0
r r r e−yi yi−1 (1 − e−yi )−1
∞
()r
n−r
i=1
i=1
d d
(18)
r r r e−yi yi−1 (1 − e−yi )−1 i=1
× 1 − (1 − e−yr )
n−r
i=1
d d:
i=1
(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.
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2.4. Bayes estimator under LINEX loss function (BELF) Following Zellner (1986), the Bayes estimators for the shape parameters and of EWD under LLF are 1 Qbl = − log (E(e−a )) a and 1 Qbl = − log(E(e−a )); a respectively, where E(·) denotes the posterior expectation. After simpli@cation, we have
j6 1 (20) Qbl = − log a j1 and
j7 1 Q ; bl = − log a j1
(21)
where j1 is given in (11), r r r 1 c ∞ r r−1 −a −yi −1 e e yi (1 − e−yi )−1 j6 = j1 0 0 i=1
× 1 − (1 − e−yr ) and 1 j7 = j1
0
c
0
∞
r
n−r
r−1 −a
e
i=1
i=1
d d
(22)
r r r −1 −yi e yi (1 − e−yi )−1 i=1
i=1
i=1
n−r × 1 − (1 − e−yr ) d d:
(23)
As mentioned earlier, the integrals involved in (21) and (22) are not solvable analytically and, therefore, the solution can be obtained using 16-point Gaussian quadrature formulas. 2.5. Generalized maximum likelihood estimator (GMLE) The GMLE of a parameter is the value of the parameter that maximizes the concerned marginal posterior density (see Martz and Waller, 1982) for details. The marginal posterior 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 for GMLEs do not exist. However, GMLEs of the parameters can be obtained with the help of numerical iterative methods without much diKculty. The results reported here are based on the procedure developed by Singh et al. (2002) which provides global maxima for the concerned posteriors. The estimators of the parameters of and are denoted by QGML and QGML , respectively.
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3. Numerical illustration Example 3.1. To illustrate the usefulness of the proposed estimators obtained in Section 2 with real situations, we considered here the real data-set initially reported by Aarset (1987) to identify the bathtub hazard rate contains lifetime of 50 devices. Mudholkar and Srivastava (1993) used this in context of three-parameter EWD to study the suitability of the model with bathtub hazard rate. Hence we obtained the proposed estimators for Aarset data and summarized it in Table 1. Bayes estimators have evaluated for the prior hyper-parameter c = 4, 10 and 12 and their corresponding values have shown in Table 1. Table 1 revealed that the Bayes estimators are not seems very sensitive with variation of ‘c’. It is also worth mentioned that though the Bayes estimators developed with non-informative prior (vague) yet the estimated values of Bayes estimators are not very far from the estimated values of MLE. Obviously, we do not expect much to conclude from this reanalysis, perhaps we are capable to show that the proposed estimators can be easily obtained in practical situations 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 only and other values can also be taken for generating the samples from EWD. In order to get a type II censored data, the generated observations were ordered and the largest four observations were removed so that the observed failures consist of @rst 6 items only. 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, (BESF and BELF with a = 1:0) are close to the true values of and as compared to MLE and GMLE. The change in the values of ‘a’ does eLect the BELF estimates only. But on the basis of a single sample estimate, it will be illogical and inappropriate to infer that BESF and BELF perform better than MLE and GMLE. One way to study the performances of these estimators would be to study their behavior for long term use. Table 1 Estimates of and (c = 4; 10; 12; n = 50) for Aarset (1987) data
Hyper-parameter
c=4
Estimators
GMLE BESF BELF (a = 1:0E − 05) BELF (a = 1:0) BELF (a = −1:0)
0.290 0.268 0.268 0.268 0.268
6.550 6.665 6.665 6.256 7.153
0.290 0.271 0.271 0.277 0.277
6.730 6.745 6.745 6.419 7.366
0.290 0.312 0.313 0.322 0.322
6.790 6.798 6.798 6.717 7.649
MLE
c = 10
0.276
c = 12
6.826
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Table 2 Estimates of and for c = 4, 10, 12, n = 10 and r=n = 0:6 for Simulated data
Hyper-parameter
c=4
Estimators
GMLE BESF BELF (a = 1:0E − 03) BELF (a = −1:0) BELF (a = 1:0)
2.752 2.460 2.460 2.581 2.034
0.119 0.406 0.406 0.450 0.378
2.767 2.522 2.522 2.614 2.107
0.176 0.418 0.418 0.459 0.401
2.767 2.528 2.528 2.619 2.108
0.176 0.421 0.421 0.461 0.400
MLE
c = 10
7.494
c = 12
0.112
In other words, we propose to study the behavior of these estimators on the basis of their 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 risks obtained under two diLerent loss functions, namely, SELF and LLF. Risks of the estimators have been estimated on the basis of 5000 randomly generated samples of size 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 EWD although in a Bayesian framework the assumed prior distributions should normally be used 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 consider uniform 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 so as to accommodate all types of failure rates like monotone, and non-monotone and also perhaps to cover the situations where complexities are expected to occur in the calculation of MLEs (see, for example, Cheng and Amin, 1983). The censoring fraction r=n, hyper-parameter c (involved only in the risks of Bayes estimators) and the shape 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 the hyper-parameter c on the risks of Bayes estimators and GMLE, it was noticed that variation 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 although most of the reported @ndings are based on all detailed evaluations. It may be further noted that the scale on y-axis varies from @gure to @gure. The @ndings are presented below, under two diLerent situations. The @rst one is the case when over- and under-estimation are considered to be of an equal importance, that is, the situation
U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523
517
Risk of Estimators of under SELF 1.00E+03
1.00E+03
1.00E+02
1.00E+02
1.00E+01
Risk
Risk
S-1 r/n=0.4; θ=0.5
1.00E+00
S-2 r/n=0.4; θ=2.0
1.00E+01 1.00E+00
1.00E-01
1.00E-01
1.00E-02 5.00E-01
1.00E+00
1.50E+00
2.00E+00
5.00E-01
2.50E+00
1.00E+00
1.50E+00
MLE BELF(1)
GMLE BELF(0)
BESF
BELF(-1)
MLE BELF(1)
S-3 r/n=0.8; θ =0.5
1.00E+02
2.00E+00
2.50E+00
α
α GMLE BELF(0)
BESF
BELF(-1)
S-4 r/n=0.8; θ =2.0
1.00E+02
1.00E+01
Risk
Risk
1.00E+01
1.00E+00
1.00E+00
1.00E-01
1.00E-01 5.00E-01
1.00E+00
1.50E+00
2.00E+00
5.00E-01
2.50E+00
1.00E+00
1.50E+00
MLE BELF(1)
GMLE BELF(0)
2.00E+00
2.50E+00
α
α BESF
BELF(-1)
MLE BELF(1)
GMLE BELF(0)
BESF
BELF(-1)
Risk of Estimators of θ under SELF 1.00E+03
1.00E+02 1.00E+01
1.00E+01
Risk
Risk
1.00E+02
1.00E+00
1.00E+00
1.00E-01
1.00E-01
1.00E-02 5.00E-01
S-6 r/n=0.4; α =2.0
1.00E+03
S-5 r/n=0.4; α=0.5
1.00E-02 1.00E+00
1.50E+00
2.00E+00
5.00E-01
2.50E+00
1.00E+00
1.50E+00
MLE
GMLE
BELF(1)
BELF(0)
BELF(-1)
1.00E+02
1.00E+01
1.00E+01
1.00E+00
1.00E+00
1.00E-01
1.00E-01
1.00E-02 5.00E-01
1.00E+00
1.50E+00
2.00E+00
MLE
GMLE
BELF(1)
BELF(0)
2.50E+00
1.00E-02 5.00E-01
1.00E+00
GMLE
BELF(1)
BELF(0)
BESF
BELF(-1)
1.50E+00
2.00E+00
2.50E+00
θ
θ MLE
2.50E+00
S-8 r/n=0.8; α=2.0
1.00E+03
1.00E+02
Risk
Risk
BESF
S-7 r/n=0.8; α =0.5
1.00E+03
2.00E+00
θ
θ
BESF
BELF(-1)
MLE
GMLE
BELF(1)
BELF(0)
BESF
BELF(-1)
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.
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U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523
Risk of Estimators of under LLF when a = 1.0 L-2 r/n=0.4; θ=2.0
L-1 r/n=0.4; θ=0.5
Risk
Risk
1.00E+34 1.00E+31 1.00E+28 1.00E+25 1.00E+22 1.00E+19 1.00E+16 1.00E+13 1.00E+10 1.00E+07 1.00E+04 1.00E+01 1.00E-02 5.00E-01
1.00E+00
1.50E+00
2.00E+00
1.00E+63 1.00E+58 1.00E+53 1.00E+48 1.00E+43 1.00E+38 1.00E+33 1.00E+28 1.00E+23 1.00E+18 1.00E+13 1.00E+08 1.00E+03 1.00E-02
2.50E+00
5.00E-01
1.00E+00
1.50E+00
α MLE 1.00E+12
GMLE
BESF
BELF
MLE
L-3 r/n=0.8; θ=0.5
1.00E+10
1.00E+06
Risk
Risk
1.00E+08
1.00E+04 1.00E+02 1.00E+00 1.00E-02 5.00E-01
1.00E+00
1.50E+00
2.00E+00
2.50E+00
1.00E+42 1.00E+38 1.00E+34 1.00E+30 1.00E+26 1.00E+22 1.00E+18 1.00E+14 1.00E+10 1.00E+06 1.00E+02 1.00E-02 5.00E-01
GMLE
2.50E+00
BESF
BELF
L-4 r/n=0.8; θ=2.0
1.00E+00
1.50E+00
α MLE
2.00E+00
α
2.00E+00
2.50E+00
α
GMLE
BESF
BELF
MLE
GMLE
BESF
BELF
Risk of Estimators of under LLF when a = -1.0 1.00E+02
L-5 r/n=0.4; θ-0.5
1.00E+01
L-6 r/n=0.4; θ=2.0
1.00E+01
Risk
Risk
1.00E+00
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
1.00E-02 5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00
α
α MLE 1.00E+01
GMLE
BESF
BELF
MLE 1.00E+00
L-7 r/n=0.8; θ=0.5
GMLE
BESF
BELF
L-8 r/n=0.8; θ=2.0
Risk
Risk
1.00E+00 1.00E-01
1.00E-01
1.00E-02 5.00E-01
1.00E+00 1.50E+00 2.00E+00 2.50E+00
1.00E-02 5.00E-01
1.00E+00
α MLE
GMLE
BESF
1.50E+00
2.00E+00
α BELF
MLE
GMLE
Fig. 2. (L-1–L-16) Risk of estimators under LLF.
BESF
BELF
2.50E+00
U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523
519
Risk of Estimators of under LLF when a = 1.0 1.00E+25
L-9 r/n=0.4; α=0.5
1.00E+22 1.00E+19 1.00E+13
Risk
Risk
1.00E+16 1.00E+10 1.00E+07 1.00E+04 1.00E+01 1.00E-02 5.00E-01
1.00E+00
MLE
1.50E+00 θ GMLE
2.00E+00
BESF
1.00E+12
1.00E+06
1.00E+06
Risk
1.00E+08
1.00E+04
2.50E+00
1.00E+02
1.00E+00
1.00E+00 1.50E+00
2.00E+00
1.00E-02 5.00E-01
2.50E+00
GMLE
BESF
BELF
1.00E+04
1.00E+02
1.00E+00
1.50E+00
θ MLE
2.00E+00
L-12 r/n=0.8; α=2.0
1.00E+10
1.00E+08
1.00E+00
1.50E+00
MLE
BELF
L-11 r/n=0.8; α=0.5
1.00E-02 5.00E-01
1.00E+00
θ
2.00E+00
2.50E+00
θ
GMLE
BESF
MLE
BELF
GMLE
BESF
BELF
Risk of Estimators of under LLF when a = 1.0 1.00E+01
L-13 r/n=0.4; α=0.5
L-14 r/n=0.4; α=2.0
1.00E+01
1.00E+00
Risk
Risk
1.00E+00
1.00E-01
1.00E-02 5.00E-01
1.00E-01
1.00E+00
1.50E+00
2.00E+00
1.00E-02 5.00E-01
2.50E+00
1.00E+00
MLE
1.50E+00
2.00E+00
2.50E+00
θ
θ GMLE
BESF
MLE
BELF
GMLE
BESF
BELF
1.00E+01 L-16 r/n=0.8; α=2.0
L-15 r/n=0.8; α=0.5
1.00E+01
1.00E+00
Risk
1.00E+00
Risk
Risk
L-10 r/n=0.4; α=2.0
5.00E-01
2.50E+00
1.00E+12 1.00E+10
1E+28 1E+25 1E+22 1E+19 1E+16 1E+13 1E+10 1E+07 10000 10 0.01
1.00E-01
1.00E-01
1.00E-02 5.00E-01
1.00E+00
1.50E+00
2.00E+00
2.50E+00
1.00E-02 5.00E-01
1.00E+00
θ MLE
GMLE
BESF
1.50E+00
2.00E+00
θ BELF
MLE
Fig. 2. continued.
GMLE
BESF
BELF
2.50E+00
520
U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523 Risk of the Various Estimators of Under LLF for a = 1.0 and a= -1.0 1.00E+00
1.00E+00
L-17 r/n=0.8; α=2.0; a=1.0 Risk
Risk
L-18 r/n=0.8; θ=2.0; a = -1.0
1.00E-01
1.00E-01 1.00E-02
1.00E-02 1.00E-03
1.00E-03
1.00E-04
1.00E-04 5.00E-01
1.00E+00
1.50E+00
2.00E+00
5.00E-01
2.50E+00
1.00E+00
1.50E+00
MLE
BESF
2.00E+00
2.50E+00
θ
θ BELF
MLE
GMLE
BESF
BELF
GMLE
Risk of the VariousEstimators of Under LLF for a = 1.0 and a = -1.0 1.00E+01
1.00E-02 1.00E-03 5.00E-01
L-20 r/n=0.8; α=2.0; a = -1.0
1.00E+00
1.00E-01
Risk
Risk
1.00E+00
L-19 r/n=0.8; α=2.0; a=1.0
1.00E-01 1.00E-02
1.00E+00
1.50E+00
2.00E+00
2.50E+00
1.00E-03 5.00E-01
1.00E+00
1.50E+00
θ MLE
BESF
2.00E+00
2.50E+00
θ BELF
GMLE
MLE
BESF
BELF
GMLE
Fig. 2. continued.
where use of symmetric loss function is justi@ed. The second one concerns with the situation where over- and under-estimation are of an unequal importance, that is, when the 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, use of symmetric loss is most justi@ed and, therefore, for comparing the performance and studying 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 hence one can agree that BESF is more justi@ed estimators than BELF. It may be interesting to 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 than those of . Risks of the estimators obtained by diLerent methods also show diLerent rate of decrements. Among the various estimators of , the rate of decrement for BELF 6 BESF ¡ MLE ¡ GMLE whereas for the estimators of the trend in rate of change of concerned risks are similar to those of except for MLE and GMLE. The rate of increase in the risk of GMLE is less than that of MLE (see Fig. 1 S-1 and S-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 of BESF and BELF early increase and then decrease (see Fig. 1 (S-1–S-4)). It has also been noted that increase in results in a slight decrease in the magnitude of the risks
U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523
521
of the estimators of but the relative performance remains the same (see Fig. 1 S-1 and 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 MLE and BELF with a = −1:0 while increment in other estimators are small in comparison to MLE and BELF with a = −1:0 (see Fig. 1(S-5–S-8)). Risks of the estimators do not vary much with variation in (see two consecutive combinations of Fig. 1 from S-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 note here that BELF with a = 1:0 have smallest risk among all, though risk of BESF is also close to it. The risk of GMLE is largest over the considered parameter space while risk of MLE is less than that of GMLE but close to those of GMLE. The diLerence between 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 found to 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 other estimators are more or less close to the risk of BELF with a = 1:0 and on the basis of the magnitude of risk, the relative position of various estimators can be expressed as BESF ¡ GMLE ¡ MLE. It is also noted that the diLerence between risk of the estimators 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 situations and, therefore, LLF will be considered here due to its properties as discussed in Section 1. It is to be mentioned again here that the nature of LLF changes according to its shape parameter a. Since ‘a’ is involved in the expression of BELF, the estimators of and obtained under BELF also change with changes in a. We, therefore, consider only the appropriate BELF estimator that matches with the de@nition of LLF. That is, when over-estimation/under-estimation becomes more serious than under-estimation/over-estimation, the risk of the estimators obtained under LLF with positive/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 with a ¿ 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 noted for risks obtained under SELF (see even and odd combinations of Fig. 2 from L-1 to L-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 increases when 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,
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risk of Bayes estimators decreases in general as increases while the risk of MLE and GMLE 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 more serious (a = 1:0); see Fig. 2 (L-9–L-12). But the circumstances where under-estimation becomes more serious (a = −1:0) the risk of all the estimators increases as increases which can be seen from Fig. 2 L-13 to L16. Comparison of various estimators: When a = 1:0; this is the case where overestimation is more serious than under-estimation. BELF of the parameters show smallest risk although the risk of BESF is near to the risk of BELF. The diLerence between the 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 more than 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 overestimation, the risk of the BESF is smallest among the risk of all other estimators while 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 for a = 1:0. Finally, to complete the study, we considered to evaluate the risks of various estimators 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 is often seen that classical MLEs may outperform the Bayes estimators for large values of n. Fig. 2 (L-17–L-20) show the risk of various estimators of and under LLF for a = 1:0 and −1:0. It is obvious that the risk of all the estimators of is very close to each other (see Fig. 2 L-17 and L-18) while for the risks of MLE and GMLE are smaller as compared to those of BESF and BELF (see Fig. 2 L-19 and L-20). So for , Bayes estimators can still be used whereas for , one should preferably consider the MLE or GMLE. 5. Conclusions The comprehensive comparison of the risks of the estimators and eLects of concerned parameters on their risks disclose that if the samples are highly censored, the BESF and BELF give smaller risks than those of MLE and GMLE. However, the risks of GMLE and MLE come closer to the risks of BELF and BESF for small censoring fractions. GMLE for always has greater risk in comparison to the MLE for all the considered 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 considered more serious than over-estimation. The BELF of with a = −1:0 under SELF shows higher 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 the estimation of as the risk of BESF and BELF are not much diLerent. However, for estimation of one can use safely either GMLE or BESF.
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Acknowledgements Authors are thankful to the co-editor and referees for their comments and suggestions that 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. References Aarset, M.V., 1987. How to identify bathtub hazard rate. IEEE Trans. Reliab. R-36, 106–108. Aitchison, J., Dunsmore, I.R., 1975. Statistical Prediction Analysis. Cambridge University Press, London. Bain, L.J., 1974. Analysis for the linear failure-rate life-testing distribution. Technometrics 16, 551–559. Berger, J.O., 1980. Statistical Decision Theory Foundations, Concept, and Methods. Springer, New York. Cheng, R.C.H., Amin, N.A.K., 1983. Estimating parameters in continuous univariate distribution with a shifted origin. J. Roy. Statist. Soc. B 45 (3), 394–403. Ferguson, T.S., 1967. Mathematical Statistics: A Decision Theoretic Approach. Academic Press, New York. Gore, A.P., Paranjape, S., Rajarshi, M.B., Gadgil, M., 1986. Some methods for summarizing survivorship data. Biometrical J. 28, 557–586. Khatree, R., 1992. Estimation of guarantee time and mean after warantee for two-parameter exponential failure model. Aust. J. Statist. 34 (2), 207–215. Lawless, J.F., 1982. Statistical Methods for Life Time Data. Wiley, New York. Martz, H.F., Waller, R.A., 1982. Bayesian Reliability Analysis. Wiley, New York. Mudholkar, G.S., Hutson, A.D., 1996. Exponentiated-Weibull family: some properties and Sood data application. Commun. Statist.-Theor. Meth. 25 (12), 3050–3083. Mudholkar, G.S., Srivastava, D.K., 1993. Exponentiated-Weibull family for analyzing bathtub failure data. IEEE Trans. Reliab. 42, 299–302. Nag, 1993. Numerical algorithms group. Mark 16, Numerical Algorithms Group, Downers Grove, IL. Parsian, A., 1990. On the admissibility of an estimator of a normal mean vector under a linex loss function. Ann. Inst. Statist. Math. 42, 657–669. Singh, U., Gupta, Pramod, K., Upadhyay, S.K., 2002. Estimation of exponentiated-Weibull shape parameters under linex loss function. Commun. Statist.- Simula. Comput. 31(4), 523–537. Varian, H.R., 1975. A Bayesian approach to real estate assessment. In: Stephen, E.F., Zellner, A. (Eds.), Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage. North-Holland, Amsterdam, pp. 195–208. Zellner, A., 1986. Bayesian estimation and prediction using asymmetric loss function. J. Amer. Statist. Assoc. 81, 446–451. Zellner, A., Geisel, M.S., 1968. Sensitivity of control to uncertainty and form of the cntenon function. In: Donald, G.W. (Ed.), The Future of Statistics. Academic Press, New York, pp. 269–289.
Estimation of parameters for exponentiated-Weibull family under type-II censoring scheme Umesh Singh, Pramod K. Gupta∗ , S.K. Upadhyay Department 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-Weibull distribution when sample is available from type-II censoring scheme. Bayes estimators have been developed under squared error loss function as well as under LINEX loss function using non-informative type of priors for the parameters. Besides, the generalized maximum likelihood estimators and the usual maximum likelihood estimators have also been attempted. It has been seen that the estimators obtained are not available in nice closed forms, although they can be easily evaluated for the given sample by using suitable numerical methods. The performance of the proposed estimators have been compared on the basis of their simulated risks (average loss over 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 Mudholkar and Hutson (1996) as a simple generalization of the well-known Weibull family by 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). c 2004 Elsevier B.V. All rights reserved. 0167-9473/$ - see front matter doi:10.1016/j.csda.2004.02.009
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distribution function of EWD are expressed as
f(y) = (1 − e−y )−1 e−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 distinguished feature of EWD from other lifetime distribution is that it accommodates nearly all types of failure rates both monotone and non-monotone (unimodal and bathtub) and includes a number of distributions as particular cases. The structural properties of EWD have been discussed by Mudholkar and Hutson (1996). It is worthwhile to mention here that the lifetime data following constant and monotone (increasing and decreasing) type of failure rates are well described by the exponential and the Weibull distributions, respectively. It may be further noted that the inferential procedures based on these models are often simple and exist in closed forms. But due to expeditious improvement in the techniques of science and technology, the non-monotone failure rate is increasingly becoming common in the @eld of engineering, medical and space exploration 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 literatures which any how serve the need when lifetime data show non-monotone failure rates. Generalized Weibull, generalized Rayleigh, generalized gamma, generalized F, mixture of Weibull distributions, lognormal and loglogistic, etc. (see Mudholkar and Hutson, 1996 for details) are few examples. But, the inferential procedures for these exempli@ed models, 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 to these distributions, EWD enjoys the advantage of being parsimonious in parameter and hence estimation of parameters of this model is expected not to pose much mathematical complexity, see, Singh et al. (2002) and may work well in the case of censoring as anticipated by Mudholkar and Hutson (1996). The estimation procedure for EWD under censoring case seems to be untouched and, therefore, we are interested to develop the estimation 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 for lifetime models are often developed using squared error loss function (SELF). No doubt, the use of SELF is well justi@ed when the loss is symmetric in nature. Its use is also very popular, perhaps, because of its mathematical simplicity. But in life testing and reliability problems, the nature of losses are not always symmetric and hence the use of SELF is forbidden and unacceptable in many situations. Inappropriateness of SELF has also been pointed out by diLerent authors. Ferguson (1967), Zellner and Geisel (1968) Aitchison and Dunsmore (1975), Varian (1975) and Berger (1980) are few among many others. It is because of this fact that Varian (1975) introduced LINEX loss function (LLF) which is the simple generalization of SELF and can be used in almost every situation. SELF can also be considered as particular case of LLF (see Zellner, 1986; Parsian, 1990; Khatree, 1992, etc.). LLF is de@ned as L( ) = b(ea − a − 1);
a = 0;
b ¿ 0;
(3)
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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 parameters of the EWD. In the next section, @rst of all the usual maximum likelihood estimators are discussed. Then the Bayes estimators under SELF as well as under the LLF are obtained. Lastly, the generalized maximum likelihood estimators for the parameters under the same Bayesian setup have also been discussed. Section 3 contains an illustrative examples based on both real and simulated data sets. The estimators obtained in Section 2 are not reducible in nice closed forms but they can be easily evaluated using suitable numerical methods. The performances of the estimators are, therefore, compared on the basis of their simulated risks obtained under both SELF and LLF separately and are summarized in the last but one section. The last section contains a brief 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 cost and time considerations, the test is terminated as soon as the rth (r 6 n) item fails. The lifetimes of these @rst r failed items say y = (y1 ; y2 ; : : : ; yr ) are observed. The likelihood function corresponding to this set-up can, therefore, be easily written as r
l(y=; ) =
r
r
i=1
i=1
n! ()r yi−1 e−yi (1 − e−yi )−1 (n − r)! i=1
× 1 − (1 − e−yr )
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=; ) = log
r
r
i=1
i=1
n! log(yi ) − yi + ( − 1) + rlog () + ( − 1) (n − r)!
r
×
log(1 − e−yi ) + (n − r)log 1 − (1 − e−yr ) :
(5)
i=1
To obtain the normal equations for the unknown parameters, we diLerentiate (5) partially with respect to the parameters and and equate to zero. The resulting equations are given below in (6) and (7), respectively, 0=
r
r
r
i=1
i=1
i=1
e−yi y log(yi ) @L r i = + log(yi ) − yi log(yi ) + ( − 1) @ (1 − e−yi )
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U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523
(n − r)(1 − e−yr )−1 e−yr yr log(yr ) ; 1 − (1 − e−yr ) r (n − r)(1 − e−yr ) log(1 − e−yr ) @L r = + : log 1 − e−yi − 0= @ 1 − (1 − e−yr ) i=1 −
(6) (7)
The solutions of above equations are the MLEs of the EWD parameters and denoted as Qml and Qml , 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 methods for @nding the numerical solutions of these equations. The solutions of these equations have been obtained using C05PCF routine of Nag (1993) which uses Powell hybrid type Newton Raphson method and provides the global solution. The routine requires all 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 1 g1 () = ; c
0 ¡ ¡ c;
(8)
1 g2 () = ;
¿ 0:
(9)
Combining (8) and (9) with Eq. (4) and using Bayes theorem, the joint posterior distribution is derived as follows: r r r 1 yi−1 (1 − e−yi )−1 (; |y) = r r−1 e−yi j1 i=1
i=1
× 1 − (1 − e−yr ) where
j1 =
c
0
0
∞
r r−1
r i=1
× 1 − (1 − e−yr )
e−yi
n−r
r
n−r
yi−1
i=1
d d:
i=1
;
(10)
r (1 − e−yi )−1 i=1
(11)
Marginal posterior of a parameter is obtained by integrating the joint posterior distribution with respect to the other parameter and hence the marginal posterior of can be written, after simpli@cation, as r r r −yi −1 (|y) = e yi j2 ; j1 i=1
i=1
(12)
U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523
where
j2 =
∞
0
r−1
r n−r (1 − e−yi )−1 1 − (1 − e−yr ) d:
513
(13)
i=1
Similarly integrating the joint posterior with respect to , the marginal posterior can be obtained as r−1 j3 (|y) = ; (14) j1 where
j3 =
c
r
0
r r r n−r e−yi yi−1 (1 − e−yi )−1 1 − (1 − e−yr ) d: i=1
i=1
(15)
i=1
2.3. Bayes estimator under squared error loss function (BESF) The Bayes estimators for parameters and of EWD may be de@ned as c Qbs = E(=y) = (=y) d; 0 ∞ Qbs = E(=y) = (=y) d; 0
respectively. These estimators can be expressed as j4 Qbs = j1
(16)
and j5 Qbs = ; j1 where j4 =
1 j1
(17)
c
0
∞
0
r+1 r−1
i=1
× 1 − (1 − e−yr ) and j5 =
1 j1
0
c
0
r r r e−yi yi−1 (1 − e−yi )−1
∞
()r
n−r
i=1
i=1
d d
(18)
r r r e−yi yi−1 (1 − e−yi )−1 i=1
× 1 − (1 − e−yr )
n−r
i=1
d d:
i=1
(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.
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U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523
2.4. Bayes estimator under LINEX loss function (BELF) Following Zellner (1986), the Bayes estimators for the shape parameters and of EWD under LLF are 1 Qbl = − log (E(e−a )) a and 1 Qbl = − log(E(e−a )); a respectively, where E(·) denotes the posterior expectation. After simpli@cation, we have
j6 1 (20) Qbl = − log a j1 and
j7 1 Q ; bl = − log a j1
(21)
where j1 is given in (11), r r r 1 c ∞ r r−1 −a −yi −1 e e yi (1 − e−yi )−1 j6 = j1 0 0 i=1
× 1 − (1 − e−yr ) and 1 j7 = j1
0
c
0
∞
r
n−r
r−1 −a
e
i=1
i=1
d d
(22)
r r r −1 −yi e yi (1 − e−yi )−1 i=1
i=1
i=1
n−r × 1 − (1 − e−yr ) d d:
(23)
As mentioned earlier, the integrals involved in (21) and (22) are not solvable analytically and, therefore, the solution can be obtained using 16-point Gaussian quadrature formulas. 2.5. Generalized maximum likelihood estimator (GMLE) The GMLE of a parameter is the value of the parameter that maximizes the concerned marginal posterior density (see Martz and Waller, 1982) for details. The marginal posterior 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 for GMLEs do not exist. However, GMLEs of the parameters can be obtained with the help of numerical iterative methods without much diKculty. The results reported here are based on the procedure developed by Singh et al. (2002) which provides global maxima for the concerned posteriors. The estimators of the parameters of and are denoted by QGML and QGML , respectively.
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3. Numerical illustration Example 3.1. To illustrate the usefulness of the proposed estimators obtained in Section 2 with real situations, we considered here the real data-set initially reported by Aarset (1987) to identify the bathtub hazard rate contains lifetime of 50 devices. Mudholkar and Srivastava (1993) used this in context of three-parameter EWD to study the suitability of the model with bathtub hazard rate. Hence we obtained the proposed estimators for Aarset data and summarized it in Table 1. Bayes estimators have evaluated for the prior hyper-parameter c = 4, 10 and 12 and their corresponding values have shown in Table 1. Table 1 revealed that the Bayes estimators are not seems very sensitive with variation of ‘c’. It is also worth mentioned that though the Bayes estimators developed with non-informative prior (vague) yet the estimated values of Bayes estimators are not very far from the estimated values of MLE. Obviously, we do not expect much to conclude from this reanalysis, perhaps we are capable to show that the proposed estimators can be easily obtained in practical situations 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 only and other values can also be taken for generating the samples from EWD. In order to get a type II censored data, the generated observations were ordered and the largest four observations were removed so that the observed failures consist of @rst 6 items only. 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, (BESF and BELF with a = 1:0) are close to the true values of and as compared to MLE and GMLE. The change in the values of ‘a’ does eLect the BELF estimates only. But on the basis of a single sample estimate, it will be illogical and inappropriate to infer that BESF and BELF perform better than MLE and GMLE. One way to study the performances of these estimators would be to study their behavior for long term use. Table 1 Estimates of and (c = 4; 10; 12; n = 50) for Aarset (1987) data
Hyper-parameter
c=4
Estimators
GMLE BESF BELF (a = 1:0E − 05) BELF (a = 1:0) BELF (a = −1:0)
0.290 0.268 0.268 0.268 0.268
6.550 6.665 6.665 6.256 7.153
0.290 0.271 0.271 0.277 0.277
6.730 6.745 6.745 6.419 7.366
0.290 0.312 0.313 0.322 0.322
6.790 6.798 6.798 6.717 7.649
MLE
c = 10
0.276
c = 12
6.826
516
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Table 2 Estimates of and for c = 4, 10, 12, n = 10 and r=n = 0:6 for Simulated data
Hyper-parameter
c=4
Estimators
GMLE BESF BELF (a = 1:0E − 03) BELF (a = −1:0) BELF (a = 1:0)
2.752 2.460 2.460 2.581 2.034
0.119 0.406 0.406 0.450 0.378
2.767 2.522 2.522 2.614 2.107
0.176 0.418 0.418 0.459 0.401
2.767 2.528 2.528 2.619 2.108
0.176 0.421 0.421 0.461 0.400
MLE
c = 10
7.494
c = 12
0.112
In other words, we propose to study the behavior of these estimators on the basis of their 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 risks obtained under two diLerent loss functions, namely, SELF and LLF. Risks of the estimators have been estimated on the basis of 5000 randomly generated samples of size 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 EWD although in a Bayesian framework the assumed prior distributions should normally be used 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 consider uniform 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 so as to accommodate all types of failure rates like monotone, and non-monotone and also perhaps to cover the situations where complexities are expected to occur in the calculation of MLEs (see, for example, Cheng and Amin, 1983). The censoring fraction r=n, hyper-parameter c (involved only in the risks of Bayes estimators) and the shape 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 the hyper-parameter c on the risks of Bayes estimators and GMLE, it was noticed that variation 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 although most of the reported @ndings are based on all detailed evaluations. It may be further noted that the scale on y-axis varies from @gure to @gure. The @ndings are presented below, under two diLerent situations. The @rst one is the case when over- and under-estimation are considered to be of an equal importance, that is, the situation
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Risk of Estimators of under SELF 1.00E+03
1.00E+03
1.00E+02
1.00E+02
1.00E+01
Risk
Risk
S-1 r/n=0.4; θ=0.5
1.00E+00
S-2 r/n=0.4; θ=2.0
1.00E+01 1.00E+00
1.00E-01
1.00E-01
1.00E-02 5.00E-01
1.00E+00
1.50E+00
2.00E+00
5.00E-01
2.50E+00
1.00E+00
1.50E+00
MLE BELF(1)
GMLE BELF(0)
BESF
BELF(-1)
MLE BELF(1)
S-3 r/n=0.8; θ =0.5
1.00E+02
2.00E+00
2.50E+00
α
α GMLE BELF(0)
BESF
BELF(-1)
S-4 r/n=0.8; θ =2.0
1.00E+02
1.00E+01
Risk
Risk
1.00E+01
1.00E+00
1.00E+00
1.00E-01
1.00E-01 5.00E-01
1.00E+00
1.50E+00
2.00E+00
5.00E-01
2.50E+00
1.00E+00
1.50E+00
MLE BELF(1)
GMLE BELF(0)
2.00E+00
2.50E+00
α
α BESF
BELF(-1)
MLE BELF(1)
GMLE BELF(0)
BESF
BELF(-1)
Risk of Estimators of θ under SELF 1.00E+03
1.00E+02 1.00E+01
1.00E+01
Risk
Risk
1.00E+02
1.00E+00
1.00E+00
1.00E-01
1.00E-01
1.00E-02 5.00E-01
S-6 r/n=0.4; α =2.0
1.00E+03
S-5 r/n=0.4; α=0.5
1.00E-02 1.00E+00
1.50E+00
2.00E+00
5.00E-01
2.50E+00
1.00E+00
1.50E+00
MLE
GMLE
BELF(1)
BELF(0)
BELF(-1)
1.00E+02
1.00E+01
1.00E+01
1.00E+00
1.00E+00
1.00E-01
1.00E-01
1.00E-02 5.00E-01
1.00E+00
1.50E+00
2.00E+00
MLE
GMLE
BELF(1)
BELF(0)
2.50E+00
1.00E-02 5.00E-01
1.00E+00
GMLE
BELF(1)
BELF(0)
BESF
BELF(-1)
1.50E+00
2.00E+00
2.50E+00
θ
θ MLE
2.50E+00
S-8 r/n=0.8; α=2.0
1.00E+03
1.00E+02
Risk
Risk
BESF
S-7 r/n=0.8; α =0.5
1.00E+03
2.00E+00
θ
θ
BESF
BELF(-1)
MLE
GMLE
BELF(1)
BELF(0)
BESF
BELF(-1)
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.
518
U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523
Risk of Estimators of under LLF when a = 1.0 L-2 r/n=0.4; θ=2.0
L-1 r/n=0.4; θ=0.5
Risk
Risk
1.00E+34 1.00E+31 1.00E+28 1.00E+25 1.00E+22 1.00E+19 1.00E+16 1.00E+13 1.00E+10 1.00E+07 1.00E+04 1.00E+01 1.00E-02 5.00E-01
1.00E+00
1.50E+00
2.00E+00
1.00E+63 1.00E+58 1.00E+53 1.00E+48 1.00E+43 1.00E+38 1.00E+33 1.00E+28 1.00E+23 1.00E+18 1.00E+13 1.00E+08 1.00E+03 1.00E-02
2.50E+00
5.00E-01
1.00E+00
1.50E+00
α MLE 1.00E+12
GMLE
BESF
BELF
MLE
L-3 r/n=0.8; θ=0.5
1.00E+10
1.00E+06
Risk
Risk
1.00E+08
1.00E+04 1.00E+02 1.00E+00 1.00E-02 5.00E-01
1.00E+00
1.50E+00
2.00E+00
2.50E+00
1.00E+42 1.00E+38 1.00E+34 1.00E+30 1.00E+26 1.00E+22 1.00E+18 1.00E+14 1.00E+10 1.00E+06 1.00E+02 1.00E-02 5.00E-01
GMLE
2.50E+00
BESF
BELF
L-4 r/n=0.8; θ=2.0
1.00E+00
1.50E+00
α MLE
2.00E+00
α
2.00E+00
2.50E+00
α
GMLE
BESF
BELF
MLE
GMLE
BESF
BELF
Risk of Estimators of under LLF when a = -1.0 1.00E+02
L-5 r/n=0.4; θ-0.5
1.00E+01
L-6 r/n=0.4; θ=2.0
1.00E+01
Risk
Risk
1.00E+00
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
1.00E-02 5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00
α
α MLE 1.00E+01
GMLE
BESF
BELF
MLE 1.00E+00
L-7 r/n=0.8; θ=0.5
GMLE
BESF
BELF
L-8 r/n=0.8; θ=2.0
Risk
Risk
1.00E+00 1.00E-01
1.00E-01
1.00E-02 5.00E-01
1.00E+00 1.50E+00 2.00E+00 2.50E+00
1.00E-02 5.00E-01
1.00E+00
α MLE
GMLE
BESF
1.50E+00
2.00E+00
α BELF
MLE
GMLE
Fig. 2. (L-1–L-16) Risk of estimators under LLF.
BESF
BELF
2.50E+00
U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523
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Risk of Estimators of under LLF when a = 1.0 1.00E+25
L-9 r/n=0.4; α=0.5
1.00E+22 1.00E+19 1.00E+13
Risk
Risk
1.00E+16 1.00E+10 1.00E+07 1.00E+04 1.00E+01 1.00E-02 5.00E-01
1.00E+00
MLE
1.50E+00 θ GMLE
2.00E+00
BESF
1.00E+12
1.00E+06
1.00E+06
Risk
1.00E+08
1.00E+04
2.50E+00
1.00E+02
1.00E+00
1.00E+00 1.50E+00
2.00E+00
1.00E-02 5.00E-01
2.50E+00
GMLE
BESF
BELF
1.00E+04
1.00E+02
1.00E+00
1.50E+00
θ MLE
2.00E+00
L-12 r/n=0.8; α=2.0
1.00E+10
1.00E+08
1.00E+00
1.50E+00
MLE
BELF
L-11 r/n=0.8; α=0.5
1.00E-02 5.00E-01
1.00E+00
θ
2.00E+00
2.50E+00
θ
GMLE
BESF
MLE
BELF
GMLE
BESF
BELF
Risk of Estimators of under LLF when a = 1.0 1.00E+01
L-13 r/n=0.4; α=0.5
L-14 r/n=0.4; α=2.0
1.00E+01
1.00E+00
Risk
Risk
1.00E+00
1.00E-01
1.00E-02 5.00E-01
1.00E-01
1.00E+00
1.50E+00
2.00E+00
1.00E-02 5.00E-01
2.50E+00
1.00E+00
MLE
1.50E+00
2.00E+00
2.50E+00
θ
θ GMLE
BESF
MLE
BELF
GMLE
BESF
BELF
1.00E+01 L-16 r/n=0.8; α=2.0
L-15 r/n=0.8; α=0.5
1.00E+01
1.00E+00
Risk
1.00E+00
Risk
Risk
L-10 r/n=0.4; α=2.0
5.00E-01
2.50E+00
1.00E+12 1.00E+10
1E+28 1E+25 1E+22 1E+19 1E+16 1E+13 1E+10 1E+07 10000 10 0.01
1.00E-01
1.00E-01
1.00E-02 5.00E-01
1.00E+00
1.50E+00
2.00E+00
2.50E+00
1.00E-02 5.00E-01
1.00E+00
θ MLE
GMLE
BESF
1.50E+00
2.00E+00
θ BELF
MLE
Fig. 2. continued.
GMLE
BESF
BELF
2.50E+00
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U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523 Risk of the Various Estimators of Under LLF for a = 1.0 and a= -1.0 1.00E+00
1.00E+00
L-17 r/n=0.8; α=2.0; a=1.0 Risk
Risk
L-18 r/n=0.8; θ=2.0; a = -1.0
1.00E-01
1.00E-01 1.00E-02
1.00E-02 1.00E-03
1.00E-03
1.00E-04
1.00E-04 5.00E-01
1.00E+00
1.50E+00
2.00E+00
5.00E-01
2.50E+00
1.00E+00
1.50E+00
MLE
BESF
2.00E+00
2.50E+00
θ
θ BELF
MLE
GMLE
BESF
BELF
GMLE
Risk of the VariousEstimators of Under LLF for a = 1.0 and a = -1.0 1.00E+01
1.00E-02 1.00E-03 5.00E-01
L-20 r/n=0.8; α=2.0; a = -1.0
1.00E+00
1.00E-01
Risk
Risk
1.00E+00
L-19 r/n=0.8; α=2.0; a=1.0
1.00E-01 1.00E-02
1.00E+00
1.50E+00
2.00E+00
2.50E+00
1.00E-03 5.00E-01
1.00E+00
1.50E+00
θ MLE
BESF
2.00E+00
2.50E+00
θ BELF
GMLE
MLE
BESF
BELF
GMLE
Fig. 2. continued.
where use of symmetric loss function is justi@ed. The second one concerns with the situation where over- and under-estimation are of an unequal importance, that is, when the 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, use of symmetric loss is most justi@ed and, therefore, for comparing the performance and studying 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 hence one can agree that BESF is more justi@ed estimators than BELF. It may be interesting to 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 than those of . Risks of the estimators obtained by diLerent methods also show diLerent rate of decrements. Among the various estimators of , the rate of decrement for BELF 6 BESF ¡ MLE ¡ GMLE whereas for the estimators of the trend in rate of change of concerned risks are similar to those of except for MLE and GMLE. The rate of increase in the risk of GMLE is less than that of MLE (see Fig. 1 S-1 and S-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 of BESF and BELF early increase and then decrease (see Fig. 1 (S-1–S-4)). It has also been noted that increase in results in a slight decrease in the magnitude of the risks
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of the estimators of but the relative performance remains the same (see Fig. 1 S-1 and 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 MLE and BELF with a = −1:0 while increment in other estimators are small in comparison to MLE and BELF with a = −1:0 (see Fig. 1(S-5–S-8)). Risks of the estimators do not vary much with variation in (see two consecutive combinations of Fig. 1 from S-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 note here that BELF with a = 1:0 have smallest risk among all, though risk of BESF is also close to it. The risk of GMLE is largest over the considered parameter space while risk of MLE is less than that of GMLE but close to those of GMLE. The diLerence between 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 found to 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 other estimators are more or less close to the risk of BELF with a = 1:0 and on the basis of the magnitude of risk, the relative position of various estimators can be expressed as BESF ¡ GMLE ¡ MLE. It is also noted that the diLerence between risk of the estimators 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 situations and, therefore, LLF will be considered here due to its properties as discussed in Section 1. It is to be mentioned again here that the nature of LLF changes according to its shape parameter a. Since ‘a’ is involved in the expression of BELF, the estimators of and obtained under BELF also change with changes in a. We, therefore, consider only the appropriate BELF estimator that matches with the de@nition of LLF. That is, when over-estimation/under-estimation becomes more serious than under-estimation/over-estimation, the risk of the estimators obtained under LLF with positive/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 with a ¿ 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 noted for risks obtained under SELF (see even and odd combinations of Fig. 2 from L-1 to L-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 increases when 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,
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U. Singh et al. / Computational Statistics & Data Analysis 48 (2005) 509 – 523
risk of Bayes estimators decreases in general as increases while the risk of MLE and GMLE 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 more serious (a = 1:0); see Fig. 2 (L-9–L-12). But the circumstances where under-estimation becomes more serious (a = −1:0) the risk of all the estimators increases as increases which can be seen from Fig. 2 L-13 to L16. Comparison of various estimators: When a = 1:0; this is the case where overestimation is more serious than under-estimation. BELF of the parameters show smallest risk although the risk of BESF is near to the risk of BELF. The diLerence between the 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 more than 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 overestimation, the risk of the BESF is smallest among the risk of all other estimators while 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 for a = 1:0. Finally, to complete the study, we considered to evaluate the risks of various estimators 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 is often seen that classical MLEs may outperform the Bayes estimators for large values of n. Fig. 2 (L-17–L-20) show the risk of various estimators of and under LLF for a = 1:0 and −1:0. It is obvious that the risk of all the estimators of is very close to each other (see Fig. 2 L-17 and L-18) while for the risks of MLE and GMLE are smaller as compared to those of BESF and BELF (see Fig. 2 L-19 and L-20). So for , Bayes estimators can still be used whereas for , one should preferably consider the MLE or GMLE. 5. Conclusions The comprehensive comparison of the risks of the estimators and eLects of concerned parameters on their risks disclose that if the samples are highly censored, the BESF and BELF give smaller risks than those of MLE and GMLE. However, the risks of GMLE and MLE come closer to the risks of BELF and BESF for small censoring fractions. GMLE for always has greater risk in comparison to the MLE for all the considered 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 considered more serious than over-estimation. The BELF of with a = −1:0 under SELF shows higher 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 the estimation of as the risk of BESF and BELF are not much diLerent. However, for estimation of one can use safely either GMLE or BESF.
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Acknowledgements Authors are thankful to the co-editor and referees for their comments and suggestions that 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. References Aarset, M.V., 1987. How to identify bathtub hazard rate. IEEE Trans. Reliab. R-36, 106–108. Aitchison, J., Dunsmore, I.R., 1975. Statistical Prediction Analysis. Cambridge University Press, London. Bain, L.J., 1974. Analysis for the linear failure-rate life-testing distribution. Technometrics 16, 551–559. Berger, J.O., 1980. Statistical Decision Theory Foundations, Concept, and Methods. Springer, New York. Cheng, R.C.H., Amin, N.A.K., 1983. Estimating parameters in continuous univariate distribution with a shifted origin. J. Roy. Statist. Soc. B 45 (3), 394–403. Ferguson, T.S., 1967. Mathematical Statistics: A Decision Theoretic Approach. Academic Press, New York. Gore, A.P., Paranjape, S., Rajarshi, M.B., Gadgil, M., 1986. Some methods for summarizing survivorship data. Biometrical J. 28, 557–586. Khatree, R., 1992. Estimation of guarantee time and mean after warantee for two-parameter exponential failure model. Aust. J. Statist. 34 (2), 207–215. Lawless, J.F., 1982. Statistical Methods for Life Time Data. Wiley, New York. Martz, H.F., Waller, R.A., 1982. Bayesian Reliability Analysis. Wiley, New York. Mudholkar, G.S., Hutson, A.D., 1996. Exponentiated-Weibull family: some properties and Sood data application. Commun. Statist.-Theor. Meth. 25 (12), 3050–3083. Mudholkar, G.S., Srivastava, D.K., 1993. Exponentiated-Weibull family for analyzing bathtub failure data. IEEE Trans. Reliab. 42, 299–302. Nag, 1993. Numerical algorithms group. Mark 16, Numerical Algorithms Group, Downers Grove, IL. Parsian, A., 1990. On the admissibility of an estimator of a normal mean vector under a linex loss function. Ann. Inst. Statist. Math. 42, 657–669. Singh, U., Gupta, Pramod, K., Upadhyay, S.K., 2002. Estimation of exponentiated-Weibull shape parameters under linex loss function. Commun. Statist.- Simula. Comput. 31(4), 523–537. Varian, H.R., 1975. A Bayesian approach to real estate assessment. In: Stephen, E.F., Zellner, A. (Eds.), Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage. North-Holland, Amsterdam, pp. 195–208. Zellner, A., 1986. Bayesian estimation and prediction using asymmetric loss function. J. Amer. Statist. Assoc. 81, 446–451. Zellner, A., Geisel, M.S., 1968. Sensitivity of control to uncertainty and form of the cntenon function. In: Donald, G.W. (Ed.), The Future of Statistics. Academic Press, New York, pp. 269–289.
