International Journal of Mathematics and Mathematical Sciences

Volume 2014 (2014), Article ID 519469, 21 pages

http://dx.doi.org/10.1155/2014/519469

Research Article

Bayesian Inference of Burr Type VIII Distribution Based on Censored Samples

Navid Feroz

Department of Mathematics and Statistics, Allama Iqbal Open University, Islamabad, Pakistan

Received 9 January 2014; Accepted 4 April 2014; Published 13 May 2014

Academic Editor: Hans Engler

Copyright © 2014 Navid Feroz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with estimation of the parameter of Burr type VIII distribution under a Bayesian framework using censored samples. The Bayes estimators and associated risks have been derived under the assumption of five priors and three loss functions. The comparison among the performance of different estimators has been made in terms of posterior risks. A simulation study has been conducted in order to assess and compare the performance of different estimators. The study proposes the use of inverse Levy prior based on quadratic loss function for Bayes estimation of the said parameter.

1. Introduction

Burr [1] introduced twelve different forms of the cumulative distribution functions of Burr distribution. Among these twelve distribution functions, Burr type X and Burr type XII have been discussed by most of the analysts, while the remaining density functions are still waiting for the attention of the researchers. Soliman [2] derived the Bayes estimates of the parameter of Burr type XII distribution relative to quadratic, LINEX, and generalized entropy loss functions. Based on Lindley’s approximation, the comparisons were made between maximum likelihood and Bayes estimation. Monte Carlo simulation was also used. Joshi [3] obtained the Bayesian estimation of reliability of generalized Burr distribution with the help of a censored sample. The precautionary and squared error loss functions were used for estimation. The noninformative and beta prior distributions were assumed for posterior analysis. Asgharzadeh and Valiollahi [4] considered the uniformly minimum variance unbiased (UMVU) of Burr model under progressively type II censored samples; the Bayes and empirical Bayes estimates for the unknown parameter and the reliability function of the Burr model were derived. The Bayes and empirical Bayes estimates were obtained on the basis of absolute error and logarithmic loss functions. A numerical example and a Monte Carlo simulation study were used to illustrate the applicability of the results. Saleem and Aslam [5] analyzed the parameter of the Rayleigh distribution under censored data. AL-Hussaini and Hussein [6] presented the maximum likelihood and Bayes estimators of the parameters, survival function (SF), and hazard rate function (HRF) for the three-parameter exponentiated Burr type XII distribution under type II censored scheme. Soliman et al. [7] dealt with Bayesian inference and prediction problems of the Burr type XII distribution based on progressive first failure censored data. Squared error loss function was proposed for estimation. Gibbs sampling procedure was applied to draw Markov chain Monte Carlo (MCMC) samples. A simulation study was carried out to compare the proposed Bayes estimators with the maximum likelihood estimators. A real life data set was also used to illustrate the results derived. Singh [8] discussed the uniformly minimum variance unbiased estimate (UMVUE) of the probability density function (pdf) of the Burr distribution. The UMVUE of the cumulative distribution function (CDF), pth quantile, and rth moment of the Burr distribution were also obtained. Feroze and Aslam [9] considered the Bayesian analysis of Gumbel type II distribution under doubly censored samples using different priors (informative and noninformative) and loss functions (symmetric and asymmetric). Bayesian credible intervals along with posterior predictive distributions were also derived.

One of the important distributions from Burr family of distributions is Burr type VIII distribution. This distribution can be used to model the lifetime data. Burr [1] has indicated that this distribution can be used as an alternative lifetime model in many cases. Its two-parametric case gives more flexibility as compared to exponential and Rayleigh distribution in order to model lifetime data. In this paper, the Bayesian inference of Burr type VIII distribution has been considered using different loss functions and priors on the basis of three censoring techniques.

The probability density function (pdf) of Burr type VIII distribution is

The cumulative distribution function (CDF) of the distribution is

2. Prior Distributions

The prior distribution is a key part of Bayesian analysis and represents the information about an uncertain parameter that is combined with the probability distribution of new data to derive the posterior distribution, which in turn is used for future inferences and decision making. Therefore, prior subject-matter knowledge about a parameter is a vital aspect of the inference process. The prior distributions can be categorized as informative and noninformative priors. As their names suggest, the informative priors are used when the prior information regarding the current experiment is available (may be in the form of prior beliefs), while when such prior information is not available the use of noninformative priors becomes inevitable. We have assumed both informative and noninformative priors for the Bayesian analysis of the said parameter.

Among noninformative priors the uniform prior proposed by Laplace [10] and Jeffreys prior suggested by Jeffreys [11] are the most famous priors. We have assumed these priors along with some informative priors to derive the posterior distributions.

The uniform prior is assumed to be

Jeffreys prior is defined to be

The informative prior for the parameter is assumed to be gamma distribution as

The informative prior for the parameter is assumed to be exponential distribution as

The informative prior for the parameter is assumed to be inverse Levy distribution as

The informative priors presented in (5)–(7) have been used in order to make our analysis more versatile. As the Burr type VIII distribution can be used to model the lifetime data, prior information available in such form can effectively be used. It should be noted that the gamma, exponential, and inverse Levy distributions can be used to model the lifetime data. So, these distributions can be used as prior distribution for the analysis of Burr type VIII distribution under Bayesian framework.

3. Bayesian Estimation under Left Censored Samples

Censoring is a useful technique when the value of a measurement or observation is only partially known. That is, it is used when an observed value is outside the range of a measuring instrument or the measure outside a range is not desired. It has a great usage in life testing experiments where continuity of the experiment till the last failure is not feasible. Censoring has many types. However, we have considered left, singly type II, and doubly type II censored samples for Bayesian inference of the parameter of Burr type VIII distribution.

Let be the last – order statistics from a sample of size from Burr type VIII distribution. Then the likelihood function for the observations is

The details for the derivation of the likelihoods function can be seen from the appendix. Now the generalized expressions for posterior distributions, Bayesian estimators, posterior risks, posterior predictive distributions, posterior predictive intervals, and credible intervals have been presented in the following. The expressions under priors given in (3), (4), (5), (6), and (7) can be derived by putting  , , , ;  , , , ;  , , , ;  , , , ; and   , , , , respectively, in the generalized results.

The generalized posterior distribution under left censored samples is

where

The details for the derivation of the posterior distribution can be seen from the appendix.

3.1. Loss Functions

The decision theory suggests that in order to select the best estimator a loss function must be specified and used to estimate the risk associated with each of the possible estimates. Since there is no definite analytical process that allows us to identify the proper loss function to be used, most of the analysts use the squared error loss function which is symmetrical and associates equal importance to the losses due to overestimation and underestimation of equal magnitude and obtain the posterior mean as Bayesian estimate. But in situations where the loss is asymmetric, we have to employ some asymmetric loss functions. So, here we have assumed some symmetric and asymmetric loss functions for the analysis of the parameter under study. The detail of these loss functions is as follows.

LINEX Loss Function (LLF). LINEX (linear exponential) loss function has been defined by Klebanov [12]. The expression of the loss function is .

For , we have . Then the Bayes estimator based on this loss function is .

Precautionary Loss Function (PLF). Norström [13] introduced an asymmetric precautionary loss function (PLF) which can be presented as , where is the Bayes estimator under this loss function.

Quadratic Loss Function (QLF). The quadratic loss function can be defined as . The Bayes estimator under QLF is given as .

Bayes estimator and corresponding risk under using LINEX loss function are, respectively, presented as

The details for the derivation of the Bayes estimates and posterior risks under LLF can be seen from the appendix.

Bayes estimator and corresponding risk under precautionary loss function are, respectively, presented as

The details for the derivation of the Bayes estimates and posterior risks under PLF can be seen from the appendix.

Bayes estimator and corresponding risk under quadratic loss function are, respectively, presented as

The details for the derivation of the Bayes estimates and posterior risks under QLF can be seen from the appendix.

4. Bayesian Estimation under Singly Type II Censored Samples

Under type II censored samples the experiment is terminated after observing some fixed percentage of observations. The likelihood function for the singly type II censored samples can be defined as follows.

Let us observe “” items for possible failure and only the first “” failure times have been observed; that is, and remaining “” items are still working. Under the assumptions that the lifetimes of the items are independently and identically distributed random variables following Burr type VIII distribution, the likelihood function for “” observations is

The generalized posterior distribution under singly type II censored samples is

where .

Bayes estimator and corresponding risk under LINEX loss function are, respectively, presented as

Bayes estimator and corresponding risk under precautionary loss function are, respectively, presented as

Bayes estimator and corresponding risk under quadratic loss function are, respectively, presented as

5. Bayesian Estimation under Doubly Type II Censored Samples

Doubly type II censoring is used when the samples are censored at two test termination points; that is, the observations below and above a particular point cannot be either observed or feasible to be observed. The likelihood function under the doubly type II censored samples can be defined as follows.

Consider a random sample of size “” from Burr type VIII distribution and let be the ordered observations that can only be observed. The remaining “” smallest observations and “” largest observations have been censored. Then the likelihood function for the type II doubly censored sample can be written as

The generalized posterior distribution under doubly type II censored samples is

where . Bayes estimator and corresponding risk under LINEX loss function are, respectively, presented as

Bayes estimator and corresponding risk under precautionary loss function are, respectively, presented as

Bayes estimator and corresponding risk under quadratic loss function are, respectively, presented as

6. Credible Intervals

Samaniego [14] has discussed that unlike the classical inference, the notion of interval estimation under Bayesian inference is simpler and manages to avoid potential conflicts with the observed data. The posterior distribution of the parameter comprises the basis for all Bayesian inferences about . The Bayesian counterpart of a confidence interval for is called a credibility interval for and is obtained from the posterior distribution by selecting an interval corresponding to the probability level desired. For example, any interval for which is the credibility interval for , where is the posterior density of and is the level of significance. The credibility interval used most often is the central one in which the limits are chosen to satisfy . Credibility intervals represent the statistician’s posterior judgment about intervals that contain with a given probability. They are clearly in harmony with the likelihood principle. So according to the above definition, the credible intervals for have been constructed under different censoring techniques as follows.

The generalized credible interval under left censored samples is

The generalized credible interval under singly type II censored samples is

The generalized credible interval under doubly type II censored samples is

7. Posterior Predictive Distributions and Intervals

There are circumstances (e.g., in regression analysis or time series) where out-of-sample (e.g., data at future time points or under different conditions and covariates) predictions are the major interest; such predictions may be in circumstances where the explanatory variates take different values to those actually observed. In clinical trials comparing the efficacy of an established therapy with a new therapy, the interest may be in the predictive probability that a new patient will benefit from the new therapy. The predictive distribution contains the information about the independent future random observation given preceding observations (data at hand). In context of Bayesian inference the predictive distribution is referred to as the posterior predictive distribution. For more illustration, see Bolstad [15], Congdon [16], and Bansal [17]. The posterior predictive distribution can be defined as

where is the posterior distribution, is density for future observation, and is the future observation given the sample information , from model (1). The posterior predictive distributions using different types of censored samples can be derived as follows.

The generalized posterior predictive distribution under left censored samples is

The generalized posterior predictive distribution under singly type II censored samples is

The generalized posterior predictive distribution under doubly type II censored samples is

The posterior predictive interval can be constructed by solving the following two equations:

where is the level of significance.

The posterior predictive intervals cannot be obtained in the closed form, so the approximate solution of the limits has been obtained by iterative procedure.

8. Simulation Study

The simulation study has been carried out for , and 150 using . As single sample may not fully describe the behavior of the estimators, the results have been replicated sufficiently and average of the results has been presented in Tables 1, 2, 3, 4, 5, 6, and 7. The performance of the different estimates has been compared in terms of magnitude of posterior risks. The posterior risks have been presented in parenthesis in the tables.

Table 1: Bayes estimators and posterior risks under uniform prior.

Left censored
Singly type II censored
Doubly type II censored

LLF
PLF
QLF
LLF
PLF
QLF
LLF
PLF
QLF

20
2.16360
2.24633
2.11116
2.19235
2.26019
2.15047
2.18079
2.25540
2.13425

(0.09240)
(0.08906)
(0.06250)
(0.07612)
(0.07236)
(0.05409)
(0.08355)
(0.07987)
(0.05465)

30
2.10850
2.18912
2.05740
2.13652
2.20263
2.09571
2.12525
2.19796
2.07990

(0.06234)
(0.06009)
(0.04167)
(0.05135)
(0.04882)
(0.03606)
(0.05637)
(0.05389)
(0.03121)

50
2.08038
2.15993
2.02996
2.10803
2.17326
2.06776
2.09691
2.16865
2.05216

(0.05368)
(0.05174)
(0.02500)
(0.04422)
(0.04204)
(0.02002)
(0.04854)
(0.04640)
(0.02223)

70
2.05895
2.13768
2.00905
2.08631
2.15087
2.04646
2.07531
2.14631
2.03102

(0.04109)
(0.03960)
(0.01786)
(0.03385)
(0.03218)
(0.01545)
(0.03715)
(0.03551)
(0.01337)

100
2.01683
2.09395
1.97424
2.04167
2.10484
2.01100
2.02041
2.05749
1.99583

(0.02499)
(0.02496)
(0.01250)
(0.02058)
(0.02029)
(0.01001)
(0.02259)
(0.02239)
(0.01111)

150
2.00807
2.08485
1.96665
2.02666
2.08937
2.00326
2.00605
2.03264
1.98815

(0.01644)
(0.01654)
(0.00833)
(0.01354)
(0.01344)
(0.00667)
(0.01486)
(0.01483)
(0.00741)

20
4.20964
4.37061
4.10762
4.26559
4.39758
4.18411
4.24309
4.38826
4.15254

(0.15309)
(0.14755)
(0.10355)
(0.12611)
(0.11989)
(0.08961)
(0.13843)
(0.13232)
(0.09054)

30
4.18678
4.34688
4.08531
4.24243
4.37371
4.16139
4.22005
4.36443
4.12999

(0.10328)
(0.09955)
(0.06903)
(0.08508)
(0.08089)
(0.05974)
(0.09339)
(0.08928)
(0.05170)

50
4.15175
4.31051
4.05113
4.20693
4.33711
4.12657
4.18474
4.32791
4.09543

(0.08894)
(0.08572)
(0.04142)
(0.07326)
(0.06965)
(0.03317)
(0.08042)
(0.07687)
(0.03683)

70
4.15016
4.30886
4.04958
4.20532
4.33545
4.12499
4.18314
4.32625
4.09387

(0.06807)
(0.06561)
(0.02959)
(0.05608)
(0.05331)
(0.02560)
(0.06155)
(0.05884)
(0.02216)

100
4.07290
4.22864
4.00663
4.12306
4.25064
4.06113
4.08013
4.15501
4.03049

(0.04140)
(0.04135)
(0.02071)
(0.03410)
(0.03362)
(0.01658)
(0.03743)
(0.03710)
(0.01841)

150
4.05520
4.21026
4.00399
4.09275
4.21940
4.04550
4.05113
4.10483
4.01498

(0.02724)
(0.02740)
(0.01380)
(0.02243)
(0.02227)
(0.01105)
(0.02462)
(0.02457)
(0.01228)

20
6.50207
6.75070
6.34449
6.58849
6.79236
6.46263
6.55374
6.77796
6.41388

(0.29086)
(0.28035)
(0.19674)
(0.23960)
(0.22779)
(0.17026)
(0.26301)
(0.25142)
(0.17202)

30
6.46677
6.71405
6.31004
6.55272
6.75548
6.42754
6.51815
6.74115
6.37905

(0.19624)
(0.18915)
(0.13116)
(0.16166)
(0.15369)
(0.11351)
(0.17745)
(0.16962)
(0.09823)

50
6.41266
6.65787
6.25724
6.49789
6.69896
6.37376
6.46361
6.68475
6.32567

(0.16898)
(0.16287)
(0.07870)
(0.13920)
(0.13234)
(0.06302)
(0.15280)
(0.14606)
(0.06998)

70
6.41020
6.65532
6.25485
6.49540
6.69639
6.37132
6.46114
6.68219
6.32325

(0.12934)
(0.12466)
(0.05621)
(0.10654)
(0.10129)
(0.04865)
(0.11695)
(0.11180)
(0.04210)

100
6.29087
6.53142
6.18852
6.36834
6.56540
6.27268
6.30203
6.41769
6.22537

(0.07867)
(0.07857)
(0.03935)
(0.06478)
(0.06387)
(0.03151)
(0.07111)
(0.07048)
(0.03497)

150
6.26353
6.50303
6.18443
6.32153
6.51714
6.24854
6.25724
6.34018
6.20141

(0.05175)
(0.05207)
(0.02622)
(0.04262)
(0.04231)
(0.02100)
(0.04678)
(0.04668)
(0.02333)

Table 2: Bayes estimators and posterior risks under Jeffreys prior.

Left censored
Singly type II censored
Doubly type II censored

LLF
PLF
QLF
LLF
PLF
QLF
LLF
PLF
QLF

20
2.15700
2.23948
2.10472
2.18567
2.25330
2.14392
2.17414
2.24852
2.12774

(0.09013)
(0.08687)
(0.06096)
(0.07425)
(0.07059)
(0.05276)
(0.08150)
(0.07791)
(0.05330)

30
2.10207
2.18245
2.05113
2.13001
2.19592
2.08932
2.11878
2.19126
2.07356

(0.06081)
(0.05861)
(0.04064)
(0.05009)
(0.04762)
(0.03517)
(0.05499)
(0.05256)
(0.03044)

50
2.07404
2.15335
2.02377
2.10161
2.16664
2.06146
2.09052
2.16204
2.04591

(0.05236)
(0.05047)
(0.02439)
(0.04313)
(0.04101)
(0.01953)
(0.04735)
(0.04526)
(0.02168)

70
2.05268
2.13117
2.00293
2.07996
2.14432
2.04022
2.06899
2.13977
2.02483

(0.04008)
(0.03863)
(0.01742)
(0.03301)
(0.03139)
(0.01507)
(0.03624)
(0.03464)
(0.01304)

100
2.01068
2.08757
1.96822
2.03545
2.09843
2.00487
2.01425
2.05122
1.98975

(0.02438)
(0.02435)
(0.01219)
(0.02007)
(0.01979)
(0.00976)
(0.02204)
(0.02184)
(0.01084)

150
2.00195
2.07850
1.96066
2.02049
2.08301
1.99716
1.99994
2.02645
1.98209

(0.01604)
(0.01613)
(0.00813)
(0.01321)
(0.01311)
(0.00651)
(0.01449)
(0.01447)
(0.00723)

20
4.19681
4.35729
4.09510
4.25259
4.38418
4.17136
4.23016
4.37488
4.13988

(0.14933)
(0.14393)
(0.10101)
(0.12301)
(0.11695)
(0.08741)
(0.13503)
(0.12907)
(0.08831)

30
4.17403
4.33363
4.07286
4.22950
4.36038
4.14870
4.20719
4.35113
4.11741

(0.10075)
(0.09711)
(0.06734)
(0.08299)
(0.07890)
(0.05827)
(0.09110)
(0.08708)
(0.05043)

50
4.13910
4.29737
4.03879
4.19411
4.32389
4.11399
4.17199
4.31472
4.08295

(0.08675)
(0.08362)
(0.04040)
(0.07146)
(0.06794)
(0.03235)
(0.07844)
(0.07499)
(0.03593)

70
4.13752
4.29573
4.03724
4.19251
4.32224
4.11242
4.17039
4.31307
4.08139

(0.06640)
(0.06400)
(0.02886)
(0.05470)
(0.05200)
(0.02497)
(0.06004)
(0.05739)
(0.02161)

100
4.06049
4.21575
3.99442
4.11049
4.23769
4.04875
4.06770
4.14235
4.01821

(0.04039)
(0.04034)
(0.02020)
(0.03326)
(0.03279)
(0.01618)
(0.03651)
(0.03618)
(0.01795)

150
4.04284
4.19743
3.99179
4.08028
4.20654
4.03317
4.03879
4.09232
4.00275

(0.02657)
(0.02673)
(0.01346)
(0.02188)
(0.02172)
(0.01078)
(0.02402)
(0.02397)
(0.01198)

20
6.48226
6.73013
6.32516
6.56842
6.77167
6.44294
6.53377
6.75730
6.39433

(0.28372
(0.27346)
(0.19191)
(0.23372)
(0.22220)
(0.16608)
(0.25655)
(0.24524)
(0.16779)

30
6.44706
6.69359
6.29081
6.53275
6.73489
6.40795
6.49829
6.72061
6.35961

(0.19142
(0.18450)
(0.12794)
(0.15768)
(0.14991)
(0.11072)
(0.17309)
(0.16546)
(0.09582)

50
6.39312
6.63758
6.23817
6.47809
6.67854
6.35434
6.44391
6.66438
6.30640

(0.16483
(0.15887)
(0.07676)
(0.13578)
(0.12909)
(0.06147)
(0.14905)
(0.14247)
(0.06826)

70
6.32657
6.56848
6.17324
6.41065
6.60902
6.28819
6.37684
6.59500
6.24075

(0.12616
(0.12160)
(0.05483)
(0.10393)
(0.09880)
(0.04745)
(0.11408)
(0.10905)
(0.04106)

100
6.14588
6.38089
6.04589
6.22157
6.41409
6.12811
6.15679
6.26978
6.08189

(0.07673
(0.07664)
(0.03838)
(0.06319)
(0.06230)
(0.03074)
(0.06936)
(0.06875)
(0.03411)

150
6.07266
6.30486
6.03308
6.12889
6.31854
6.05813
6.06656
6.14697
6.01243

(0.05048
(0.05079)
(0.02558)
(0.04158)
(0.04127)
(0.02048)
(0.04563)
(0.04554)
(0.02275)

Table 3: Bayes estimators and posterior risks under exponential prior.

Left censored
Singly type II censored
Doubly type II censored

LLF
PLF
QLF
LLF
PLF
QLF
LLF
PLF
QLF

20
2.13477
2.21640
2.08304
2.16315
2.23008
2.12182
2.15174
2.22535
2.10582

(0.08792)
(0.08474)
(0.05947)
(0.07242)
(0.06885)
(0.05146)
(0.07950)
(0.07599)
(0.05199)

30
2.08041
2.15996
2.02999
2.10806
2.17329
2.06779
2.09694
2.16868
2.05219

(0.05931)
(0.05717)
(0.03964)
(0.04886)
(0.04645)
(0.03431)
(0.05363)
(0.05127)
(0.02969)

50
2.05267
2.13116
2.00292
2.07995
2.14431
2.04022
2.06898
2.13976
2.02482

(0.05108)
(0.04923)
(0.02379)
(0.04207)
(0.04000)
(0.01905)
(0.04618)
(0.04415)
(0.02115)

70
2.03152
2.10920
1.98229
2.05852
2.12222
2.01920
2.04766
2.11772
2.00397

(0.03909)
(0.03768)
(0.01699)
(0.03220)
(0.03062)
(0.01470)
(0.03535)
(0.03379)
(0.01272)

100
1.98996
2.06606
1.94794
2.01447
2.07681
1.98421
1.99350
2.03008
1.96924

(0.02378)
(0.02375)
(0.01189)
(0.01958)
(0.01931)
(0.00952)
(0.02149)
(0.02130)
(0.01057)

150
1.98132
2.05708
1.94045
1.99966
2.06154
1.97658
1.97933
2.00556
1.96167

(0.01564)
(0.01574)
(0.00793)
(0.01288)
(0.01279)
(0.00635)
(0.01414)
(0.01411)
(0.00705)

20
4.15357
4.31239
4.05290
4.20877
4.33901
4.12837
4.18657
4.32980
4.09722

(0.14566)
(0.14039)
(0.09852)
(0.11999)
(0.11407)
(0.08526)
(0.13171)
(0.12590)
(0.08614)

30
4.13101
4.28898
4.03089
4.18592
4.31544
4.10595
4.16384
4.30629
4.07498

(0.09827)
(0.09472)
(0.06568)
(0.08095)
(0.07696)
(0.05684)
(0.08886)
(0.08494)
(0.04919)

50
4.09645
4.25309
3.99717
4.15089
4.27934
4.07160
4.12900
4.27026
4.04088

(0.08462)
(0.08156)
(0.03941)
(0.06971)
(0.06627)
(0.03156)
(0.07652)
(0.07314)
(0.03504)

70
4.09488
4.25146
3.99564
4.14930
4.27770
4.07004
4.12742
4.26862
4.03933

(0.06477)
(0.06243)
(0.02815)
(0.05335)
(0.05072)
(0.02436)
(0.05857)
(0.05598)
(0.02108)

100
4.01865
4.17231
3.95326
4.06814
4.19402
4.00703
4.02578
4.09966
3.97680

)

(0.03939)
(0.03935)
(0.01970)
(0.03244)
(0.03198)
(0.01578)
(0.03561)
(0.03530)
(0.01751)

150
4.00118
4.15418
3.95066
4.03824
4.16319
3.99161
3.99717
4.05015
3.96150

(0.02592)
(0.02607)
(0.01313)
(0.02134)
(0.02119)
(0.01051)
(0.02343)
(0.02338)
(0.01168)

20
6.41546
6.66078
6.25998
6.50073
6.70189
6.37654
6.46644
6.68767
6.32844

(0.27675)
(0.26675)
(0.18720)
(0.22798)
(0.21674)
(0.16200)
(0.25025)
(0.23922)
(0.16367)

30
6.38063
6.62461
6.22599
6.46543
6.66549
6.34192
6.43132
6.65135
6.29407

(0.18672)
(0.17997)
(0.12480)
(0.15381)
(0.14623)
(0.10800)
(0.16884)
(0.16139)
(0.09346)

50
6.32724
6.56918
6.17389
6.41133
6.60972
6.28886
6.37751
6.59570
6.24141

(0.16078)
(0.15497)
(0.07488)
(0.13244)
(0.12592)
(0.05996)
(0.14538)
(0.13897)
(0.06658)

70
6.26137
6.50080
6.10962
6.34459
6.54092
6.22339
6.31113
6.52704
6.17644

(0.12306)
(0.11861)
(0.05348)
(0.10137)
(0.09638)
(0.04629)
(0.11128)
(0.10637)
(0.04006)

100
6.08255
6.31513
5.98359
6.15746
6.34799
6.06497
6.09335
6.20517
6.01921

(0.07485)
(0.07476)
(0.03744)
(0.06164)
(0.06077)
(0.02998)
(0.06766)
(0.06706)
(0.03328)

150
6.01008
6.23989
5.97091
6.06574
6.25343
5.99570
6.00405
6.08363
5.95047

(0.04924)
(0.04954)
(0.02495)
(0.04055)
(0.04025)
(0.01998)
(0.04451)
(0.04442)
(0.02219)

Table 4: Bayes estimators and posterior risks under gamma prior.

Left censored
Singly type II censored
Doubly type II censored

LLF
PLF
QLF
LLF
PLF
QLF
LLF
PLF
QLF

20
2.17527
2.25845
2.12255
2.20418
2.27239
2.16207
2.19255
2.26757
2.14576

(0.07926)
(0.07640)
(0.05361)
(0.06529)
(0.06207)
(0.04640)
(0.07167)
(0.06851)
(0.04688)

30
2.11988
2.20094
2.06850
2.14805
2.21452
2.10702
2.13672
2.20982
2.09112

(0.05347)
(0.05154)
(0.03574)
(0.04405)
(0.04188)
(0.03093)
(0.04835)
(0.04622)
(0.02677)

50
2.09161
2.17159
2.04091
2.11941
2.18499
2.07892
2.10823
2.18035
2.06323

(0.04605)
(0.04438)
(0.02144)
(0.03793)
(0.03606)
(0.01717)
(0.04164)
(0.03980)
(0.01907)

70
2.07006
2.14922
2.01989
2.09757
2.16248
2.05750
2.08651
2.15789
2.04198

(0.03524)
(0.03397)
(0.01532)
(0.02903)
(0.02760)
(0.01326)
(0.03187)
(0.03046)
(0.01147)

100
2.02771
2.10525
1.98489
2.05268
2.11620
2.02185
2.03131
2.06859
2.00660

(0.02144)
(0.02141)
(0.01072)
(0.01765)
(0.01740)
(0.00859)
(0.01938)
(0.01921)
(0.00953)

150
2.01890
2.09610
1.97726
2.03760
2.10065
2.01407
2.01687
2.04361
1.99888

(0.01410)
(0.01419)
(0.00715)
(0.01161)
(0.01153)
(0.00572)
(0.01275)
(0.01272)
(0.00636)

20
4.23236
4.39420
4.12978
4.28861
4.42131
4.20668
4.26599
4.41194
4.17495

(0.13132)
(0.12657)
(0.08882)
(0.10817)
(0.10284)
(0.07687)
(0.11874)
(0.11351)
(0.07766)

30
4.20938
4.37034
4.10736
4.26532
4.39731
4.18384
4.24282
4.38798
4.15228

(0.08860)
(0.08539)
(0.05922)
(0.07298)
(0.06938)
(0.05125)
(0.08011)
(0.07658)
(0.04435)

50
4.17416
4.33377
4.07299
4.22963
4.36051
4.14883
4.20732
4.35126
4.11753

(0.07629)
(0.07353)
(0.03553)
(0.06284)
(0.05975)
(0.02845)
(0.06898)
(0.06594)
(0.03159)

70
4.17256
4.33211
4.07143
4.22801
4.35884
4.14725
4.20571
4.34960
4.11596

(0.05839)
(0.05628)
(0.02538)
(0.04810)
(0.04573)
(0.02196)
(0.05280)
(0.05047)
(0.01901)

100
4.09488
4.25146
4.02825
4.14531
4.27358
4.08304
4.10215
4.17743
4.05224

(0.03552)
(0.03547)
(0.01776)
(0.02925)
(0.02884)
(0.01423)
(0.03210)
(0.03182)
(0.01579)

150
4.07708
4.23298
4.02560
4.11484
4.24217
4.06733
4.07299
4.12698
4.03665

(0.02336)
(0.02351)
(0.01184)
(0.01924)
(0.01910)
(0.00948)
(0.02112)
(0.02108)
(0.01053)

20
6.53716
6.78713
6.37873
6.62405
6.82902
6.49750
6.58910
6.81453
6.44849

(0.24950)
(0.24048)
(0.16877)
(0.20553)
(0.19540)
(0.14605)
(0.22561)
(0.21566)
(0.14756)

30
6.50166
6.75028
6.34409
6.58808
6.79193
6.46222
6.55332
6.77753
6.41347

(0.16833)
(0.16225)
(0.11251)
(0.13867)
(0.13183)
(0.09737)
(0.15221)
(0.14550)
(0.08426)

50
6.44726
6.69379
6.29101
6.53295
6.73511
6.40815
6.49849
6.72082
6.35981

(0.14495)
(0.13971)
(0.06751)
(0.11940)
(0.11352)
(0.05406)
(0.13107)
(0.12529)
(0.06003)

70
6.38015
6.62412
6.22552
6.46495
6.66500
6.34145
6.43084
6.65086
6.29360

(0.11094)
(0.10693)
(0.04822)
(0.09139)
(0.08689)
(0.04173)
(0.10032)
(0.09590)
(0.03611)

100
6.19793
6.43493
6.09709
6.27426
6.46841
6.18002
6.20893
6.32288
6.13340

(0.06748)
(0.06740)
(0.03375)
(0.05557)
(0.05479)
(0.02703)
(0.06100)
(0.06046)
(0.03000)

150
6.12409
6.35826
6.08418
6.18080
6.37206
6.10943
6.11794
6.19903
6.06335

(0.04439)
(0.04466)
(0.02249)
(0.03656)
(0.03629)
(0.01801)
(0.04013)
(0.04004)
(0.02001)

Table 5: Bayes estimators and posterior risks under inverse Levy prior.

Left censored
Singly type II censored
Doubly type II censored

LLF
PLF
QLF
LLF
PLF
QLF
LLF
PLF
QLF

20
2.15285
2.23518
2.10068
2.18147
2.24897
2.13980
2.16996
2.24420
2.12365

(0.07731)
(0.07452)
(0.05230)
(0.06369)
(0.06055)
(0.04526)
(0.06991)
(0.06683)
(0.04572)

30
2.09803
2.17826
2.04718
2.12592
2.19170
2.08531
2.11470
2.18705
2.06957

(0.05216)
(0.05028)
(0.03486)
(0.04297)
(0.04085)
(0.03017)
(0.04717)
(0.04509)
(0.026110)

50
2.07005
2.14921
2.01988
2.09757
2.16247
2.05750
2.08650
2.15788
2.04197

(0.04492)
(0.04329)
(0.02092)
(0.03700)
(0.03518)
(0.01675)
(0.04061)
(0.03882)
(0.01860)

70
2.04873
2.12707
1.99908
2.07596
2.14020
2.03630
2.06501
2.13566
2.02094

(0.03438)
(0.03314)
(0.01494)
(0.02832)
(0.02692)
(0.01293)
(0.03109)
(0.02972)
(0.01119)

100
2.00682
2.08355
1.96444
2.03153
2.09440
2.00102
2.01038
2.04728
1.98592

(0.02091)
(0.02088)
(0.01046)
(0.01722)
(0.01698)
(0.00838)
(0.01890)
(0.01873)
(0.00930)

150
2.00673
2.08346
1.96534
2.02531
2.08798
2.00192
2.00471
2.03128
1.98682

(0.01376)
(0.01384)
(0.00697)
(0.01133)
(0.01125)
(0.00558)
(0.01243)
(0.01241)
(0.00620)

20
4.18874
4.34891
4.08723
4.24442
4.37575
4.16333
4.22203
4.36647
4.13193

(0.12809)
(0.12346)
(0.08664)
(0.10552)
(0.10032)
(0.07498)
(0.11583)
(0.11072)
(0.07575)

30
4.16600
4.32530
4.06503
4.22137
4.35199
4.14073
4.19910
4.34276
4.10949

(0.08642)
(0.08330)
(0.05776)
(0.07119)
(0.06768)
(0.04999)
(0.07814)
(0.07470)
(0.04326)

50
4.13114
4.28911
4.03102
4.18605
4.31558
4.10608
4.16397
4.30643
4.07510

(0.07441)
(0.07173)
(0.03466)
(0.06130)
(0.05828)
(0.02775)
(0.06729)
(0.06432)
(0.03082)

70
4.12956
4.28747
4.02948
4.18445
4.31393
4.10451
4.16237
4.30478
4.07354

(0.05696)
(0.05490)
(0.02475)
(0.04692)
(0.04461)
(0.02142)
(0.05150)
(0.04923)
(0.01854)

100
4.05268
4.20765
4.02703
4.10259
4.22954
4.07060
4.05988
4.13439
4.03989

(0.03464)
(0.03460)
(0.01733)
(0.02853)
(0.02813)
(0.01388)
(0.03132)
(0.03104)
(0.01540)

150
4.05250
4.20746
4.00132
4.09003
4.21659
4.04280
4.04843
4.10209
4.01231

(0.02279)
(0.02293)
(0.01155)
(0.01877)
(0.01863)
(0.00925)
(0.02060)
(0.02056)
(0.01027)

20
6.46980
6.71719
6.31300
6.55579
6.75865
6.43055
6.52120
6.74431
6.38204

(0.24337)
(0.23458)
(0.16462)
(0.20048)
(0.19060)
(0.14246)
(0.22007)
(0.21037)
(0.14393)

30
6.43467
6.68072
6.27872
6.52019
6.72195
6.39563
6.48579
6.70769
6.34738

(0.16420)
(0.15826)
(0.10975)
(0.13526)
(0.12859)
(0.09498)
(0.14847)
(0.14193)
(0.08219)

50
6.38083
6.62482
6.22618
6.46563
6.66570
6.34212
6.43153
6.65156
6.29427

(0.14139)
(0.13628)
(0.06585)
(0.11647)
(0.11073)
(0.05273)
(0.12785)
(0.12221)
(0.05855)

70
6.31440
6.55586
6.16137
6.39833
6.59632
6.27610
6.36458
6.58232
6.22875

(0.10822)
(0.10431)
(0.04703)
(0.08915)
(0.08475)
(0.04070)
(0.09786)
(0.09354)
(0.03522)

100
6.13406
6.36862
6.09523
6.20961
6.40175
6.16118
6.14495
6.25773
6.11471

(0.06582)
(0.06574)
(0.03292)
(0.05421)
(0.05344)
(0.02637)
(0.05950)
(0.05897)
(0.02926)

150
6.08716
6.31992
6.04748
6.14353
6.33363
6.07259
6.08105
6.16165
6.02679

(0.04330)
(0.04356)
(0.02194)
(0.03566)
(0.03540)
(0.01757)
(0.03914)
(0.03906)
(0.01952)

Table 6: 95% credible intervals under different priors and censoring techniques for .

Left censored
Singly type II censored
Doubly type II censored

LL
UL

LL
UL

LL
UL

Uniform prior

20
1.56781
2.75938
1.19157
1.73877
2.84372
1.10495
1.73962
2.90945
1.16983

30
1.61913
2.59787
0.97874
1.79641
2.70600
0.90960
1.78866
2.72517
0.93650

50
1.62627
2.53449
0.90822
1.80163
2.66082
0.85919
1.79136
2.66332
0.87196

70
1.66166
2.45624
0.79458
1.83777
2.58974
0.75197
1.82279
2.59834
0.77556

100
1.70699
2.32667
0.61968
1.85560
2.41714
0.56155
1.83423
2.40771
0.57348

150
1.75676
2.25937
0.50262
1.88471
2.34154
0.45683
1.85867
2.33087
0.47219

Jeffreys prior

20
1.56858
2.74542
1.17685
1.73708
2.80267
1.06559
1.73746
2.86972
1.13226

30
1.61875
2.58540
0.96664
1.79389
2.66752
0.87362
1.78579
2.68849
0.90270

50
1.62554
2.52254
0.89700
1.79889
2.62316
0.82427
1.78830
2.62765
0.83935

70
1.66030
2.44505
0.78476
1.83458
2.55341
0.71884
1.79885
2.54065
0.74180

100
1.70467
2.31670
0.61202
1.85182
2.38369
0.53187
1.79008
2.33228
0.54221

150
1.75374
2.25015
0.49640
1.88049
2.30946
0.42898
1.79945
2.24150
0.44204

Exponential prior

20
1.55362
2.71593
1.16230
1.71997
2.74490
1.02493
1.72025
2.78576
1.06552

30
1.60306
2.55776
0.95470
1.77605
2.61266
0.83660
1.76796
2.61904
0.85109

50
1.60971
2.49562
0.88591
1.78096
2.56925
0.78830
1.77039
2.64202
0.87163

70
1.64399
2.41905
0.77506
1.81620
2.50101
0.68482
1.78077
2.49203
0.71126

100
1.68773
2.29219
0.60446
1.83315
2.33487
0.50173
1.77199
2.31768
0.54570

150
1.73618
2.22645
0.49027
1.86144
2.26223
0.40079
1.78120
2.25684
0.47564

Gamma prior

20
1.62347
2.72707
1.10360
1.77884
2.78763
1.00880
1.77576
2.78477
1.00901

30
1.66664
2.57312
0.90648
1.83130
2.65756
0.82626
1.82029
2.62195
0.80166

50
1.67102
2.51219
0.84117
1.83474
2.61473
0.77998
1.82142
2.64617
0.82475

70
1.70210
2.43802
0.73592
1.86815
2.54765
0.67951
1.82981
2.49792
0.66811

100
1.74075
2.31468
0.57393
1.88157
2.38174
0.50017
1.81750
2.32602
0.50852

150
1.78615
2.25166
0.46551
1.90782
2.31004
0.40222
1.82464
2.26706
0.44242

Inverse Levy prior

20
1.60787
2.69784
1.08996
1.76125
2.73853
0.97728
1.75811
2.70183
0.94372

30
1.65039
2.54567
0.89528
1.81304
2.61087
0.79783
1.80206
2.54396
0.74190

50
1.65467
2.48544
0.83077
1.81640
2.56882
0.75242
1.80314
2.56750
0.76436

70
1.68532
2.41214
0.72682
1.84939
2.50299
0.65360
1.81138
2.42371
0.61233

100
1.72340
2.29024
0.56684
1.86256
2.34007
0.47751
1.79911
2.25700
0.45789

150
1.77685
2.23660
0.45976
1.89728
2.27883
0.38156
1.81447
2.20874
0.39427

Table 7: 95% posterior predictive intervals under different priors and censoring techniques for .

Left censored
Singly type II censored
Doubly type II censored

LL
UL

LL
UL

LL
UL

Uniform prior

20
0.49523
13.47175
12.97652
0.54924
13.88351
13.33428
0.54950
14.20442
13.65491

30
0.51144
12.68323
12.17179
0.56744
13.21114
12.64370
0.56500
13.30473
12.73974

50
0.51370
12.37380
11.86010
0.56909
12.99057
12.42147
0.56585
13.00277
12.43692

70
0.52488
11.99177
11.46689
0.58051
12.64354
12.06303
0.57578
12.68553
12.10975

100
0.53920
11.35919
10.81999
0.58614
11.80088
11.21474
0.57939
11.75484
11.17545

150
0.55492
11.03062
10.47570
0.59534
11.43179
10.83645
0.58711
11.37969
10.79258

Jeffreys prior

20
0.49548
13.40360
12.90812
0.54870
13.68310
13.13440
0.54882
14.01045
13.46163

30
0.51132
12.62235
12.11103
0.56665
13.02328
12.45663
0.56409
13.12565
12.56157

50
0.51347
12.31546
11.80199
0.56823
12.80670
12.23848
0.56488
12.82862
12.26374

70
0.52445
11.93714
11.41269
0.57950
12.46617
11.88667
0.56821
12.40388
11.83566

100
0.53846
11.31051
10.77205
0.58495
11.63757
11.05262
0.56544
11.38658
10.82113

150
0.55396
10.98561
10.43164
0.59400
11.27517
10.68117
0.56840
10.94338
10.37497

Exponential prior

20
0.49075
13.25962
12.76887
0.54330
13.40106
12.85776
0.54339
13.60054
13.05716

30
0.50637
12.48741
11.98104
0.56101
12.75544
12.19443
0.55846
12.78659
12.22813

50
0.50847
12.18403
11.67556
0.56256
12.54351
11.98094
0.55922
12.89878
12.33956

70
0.51930
11.81020
11.29091
0.57369
12.21035
11.63665
0.56250
12.16650
11.60400

100
0.53311
11.19085
10.65774
0.57905
11.39922
10.82017
0.55973
11.31530
10.75557

150
0.54842
10.86990
10.32148
0.58798
11.04458
10.45660
0.56264
11.01827
10.45563

Gamma prior

20
0.51282
13.31401
12.80119
0.56189
13.60967
13.04778
0.56092
13.59571
13.03479

30
0.52645
12.56240
12.03595
0.57846
12.97465
12.39619
0.57499
12.80080
12.22581

50
0.52784
12.26493
11.73709
0.57955
12.76555
12.18600
0.57534
12.91904
12.34370

70
0.53765
11.90282
11.36517
0.59010
12.43805
11.84795
0.57799
12.19526
11.61727

100
0.54986
11.30065
10.75079
0.59434
11.62805
11.03371
0.57411
11.35602
10.78191

150
0.56420
10.99298
10.42878
0.60264
11.27800
10.67536
0.57636
11.06816
10.49180

Inverse Levy prior

20
0.50789
13.17130
12.66342
0.55634
13.36996
12.81362
0.55535
13.19078
12.63544

30
0.52132
12.42838
11.90706
0.57270
12.74670
12.17400
0.56923
12.42004
11.85081

50
0.52267
12.13433
11.61166
0.57376
12.54141
11.96765
0.56957
12.53496
11.96539

70
0.53235
11.77647
11.24412
0.58418
12.22001
11.63583
0.57217
11.83295
11.26078

100
0.54438
11.18133
10.63695
0.58834
11.42461
10.83627
0.56830
11.01905
10.45075

150
0.56126
10.91945
10.35819
0.59931
11.12563
10.52632
0.57315
10.78344
10.21029

It is immediate from Tables 1, 2, 3, 4, 5, 6, and 7 that estimated value of the parameter converges to the true value by increasing the sample size. The parameter is overestimated for the majority of the cases. A greater tendency of overestimation is observed for larger values of the parameter. This pattern is similar under each prior, loss function, and censoring scheme. The convergence of the estimated values of the parameter under informative priors is better than those under noninformative priors. In case of loss functions the convergence is better under LLF for smaller values of the parameter, while for larger values of the parameter the convergence is faster under QLF. As far as the censoring techniques are concerned, the better convergence is observed under left censored samples for LLF and PLF, while in case of QLF the estimates under singly type II censored samples have comparatively rapid convergence.

The study also depicts that, in terms of posterior risks, the performance of the QLF is the best among all loss functions. The performance of each loss function has been negatively affected by the increasing values of the parameter. The magnitude values of risks associated with estimates using PLF and LLF are close to each other especially in large samples. So, we can replace these two loss functions by each other in the future studies. In comparison of priors, it has been observed that the informative priors have smaller risks than noninformative priors. The estimates based on inverse Levy prior are having the least amounts of posterior risks. The larger choice of the true parametric values has negatively affected the performance of each prior. The increased censoring rate results in slower convergence of the estimates and the inflated posterior risks. It is a natural consequence of the censoring that can be prevented by increased sample size. It is interesting to note that, for each prior and loss function, the posterior risks under singly type II censoring are smaller than those under other censoring techniques.

The Bayesian interval estimation exhibited a similar pattern as observed in case of point estimation. All the credible intervals are skewed toward the right. The credible intervals under informative priors are more specific than those under noninformative priors on the basis of each censoring technique. The minimum widths of credible intervals have been observed under inverse Levy prior. In addition, the credible intervals under singly type II censored samples have smaller widths as compared to other censoring techniques. Similar trends have been observed as far as the posterior predictive intervals are concerned.

9. Real Life Example

This section covers the analysis of real life data set regarding the breaking strengths of 64 single carbon fibers of length 10, presented by Lawless [18]. The idea has been to see whether the results and properties of the Bayes estimators, explored by simulation study, are applicable to a real life situation. The amounts of posterior risks associated with each estimate have been presented in parenthesis in Tables 8, 9, and 10.

Table 8: Bayes estimates and posterior risks under real life data.

Priors
Left censored
Singly type II censored
Doubly type II censored

LLF
PLF
QLF
LLF
PLF
QLF
LLF
PLF
QLF

10% censored

Uniform
47.24555
48.72147
46.64106
47.87348
49.02215
47.50956
47.62094
48.91817
47.15113

(3.30822)
(2.49507)
(0.01786)
(2.86304)
(2.28683)
(0.01724)
(3.01607)
(2.43653)
(0.01760)

Jeffreys
46.66228
47.87151
46.01342
47.33033
48.21597
46.83223
47.03304
48.06477
46.51663

(3.26701)
(2.48894)
(0.01763)
(2.82738)
(2.28121)
(0.01702)
(2.97850)
(2.43055)
(0.01738)

Exponential
46.08621
47.03637
45.39423
46.79335
47.42304
46.16456
46.45240
47.22627
45.89067

(3.21329)
(2.42312)
(0.01734)
(2.78088)
(2.22089)
(0.01674)
(2.92952)
(2.36627)
(0.01709)

Gamma
46.40313
47.35982
45.70638
47.11513
47.74915
46.48201
46.77183
47.55102
46.20624

(2.93552)
(2.33481)
(0.01585)
(2.54049)
(2.13995)
(0.01530)
(2.67628)
(2.28004)
(0.01561)

Inverse Levy
46.53661
47.52780
45.83237
47.25066
47.91850
46.61013
46.90637
47.71967
46.33360

(2.88725)
(2.29642)
(0.01558)
(2.49872)
(2.10476)
(0.01505)
(2.63227)
(2.24255)
(0.01536)

20% censored

Uniform
48.88463
50.33114
48.34057
49.75404
50.92134
49.35048
49.40203
50.34688
48.98630

(3.40898)
(2.57077)
(0.01848)
(2.94916)
(2.37172)
(0.01785)
(3.12592)
(2.50748)
(0.01828)

Jeffreys
47.86867
49.27538
47.14291
48.74232
49.75933
48.22909
48.59545
49.44202
48.29985

(3.34670)
(2.56094)
(0.01805)
(2.90939)
(2.35381)
(0.01749)
(3.04704)
(2.49551)
(0.01804)

Exponential
47.13807
48.35455
46.45451
48.09022
48.68251
47.41832
47.50274
48.04644
47.55169

(3.28574)
(2.48695)
(0.01766)
(2.85620)
(2.27935)
(0.01708)
(2.99525)
(2.40593)
(0.01771)

Gamma
47.42154
48.43907
46.48440
48.00535
48.93807
47.30361
47.70787
48.24894
47.67193

(2.99705)
(2.38132)
(0.01610)
(2.58490)
(2.19134)
(0.01556)
(2.72857)
(2.30718)
(0.01590)

Inverse Levy
47.22389
48.27313
46.52757
47.91474
48.90824
47.34369
47.79353
48.24738
46.99236

(2.91876)
(2.32932)
(0.01579)
(2.52577)
(2.13919)
(0.01522)
(2.66669)
(2.26701)
(0.01553)

Table 9: 95% credible intervals under real life data.

Priors
Left censored
Singly type II censored
Doubly type II censored

LL
UL

LL
UL

LL
UL

10% censored

Uniform
43.68060
50.81050
7.12990
44.55706
51.18990
6.63284
44.21704
51.02484
6.80780

Jeffreys
43.11961
50.20496
7.08535
44.03463
50.62604
6.59140
43.65041
50.41568
6.76526

Exponential
42.57279
49.59964
7.02685
43.52486
50.06184
6.53698
43.09770
49.80711
6.70941

Gamma
43.04499
49.76127
6.71627
43.99110
50.23916
6.24806
43.56540
49.97826
6.41286

Inverse Levy
43.20620
49.86702
6.66082
44.15242
50.34889
6.19647
43.72641
50.08633
6.35992

20% censored

Uniform
42.72716
52.83739
10.11023
43.22736
52.78174
9.55438
43.26458
52.76883
9.50425

Jeffreys
42.25671
52.09809
9.84138
42.72587
52.09732
9.37145
42.83694
51.57464
8.73770

Exponential
41.89867
51.40512
9.50645
42.34254
51.34485
9.00231
42.41062
50.82486
8.41424

Gamma
42.50686
51.06047
8.55360
42.91141
51.31689
8.40548
42.89212
50.76187
7.86975

Inverse Levy
42.74227
50.65512
7.91285
43.63056
51.40093
7.77038
43.07501
50.82409
7.74908

Table 10: 95% posterior predictive intervals under real life data.

Priors
Left censored
Singly type II censored
Doubly type II censored

LL
UL

LL
UL

LL
UL

10% censored

Uniform
13.79766
248.06530
234.26764
14.07452
249.91759
235.84308
13.96711
249.11174
235.14463

Jeffreys
13.62046
245.10895
231.48849
13.90949
247.16474
233.25524
13.78813
246.13772
232.34960

Exponential
13.44773
242.15368
228.70595
13.74847
244.41022
230.66175
13.61354
243.16658
229.55304

Gamma
13.59689
242.94278
229.34589
13.89574
245.27592
231.38018
13.76128
244.00217
230.24089

Inverse Levy
13.64781
243.45907
229.81126
13.94670
245.81164
231.86494
13.81213
244.52978
230.71765

20% censored

Uniform
13.49650
257.96091
244.46442
13.65450
257.68922
244.03472
13.66625
257.62619
243.95994

Jeffreys
13.34789
254.35153
241.00364
13.49609
254.34777
240.85168
13.53117
251.79596
238.26479

Exponential
13.23479
250.96833
237.73354
13.37500
250.67409
237.29908
13.39651
248.13541
234.73890

Gamma
13.42691
249.28570
235.85879
13.55470
250.53758
236.98288
13.54860
247.82788
234.27928

Inverse Levy
13.50127
247.30671
233.80544
13.78186
250.94788
237.16602
13.60637
248.13165
234.52528

The results regarding the analysis of the real life data displayed the characteristics of the estimates parallel to those observed under simulation study. The inverse Levy prior has been found to be the most suitable prior and QLF the best working loss function. The results under real life data also indicated that the estimates under singly type II censored samples are more efficient under each prior and loss function. So, the information lost from the right end may not be as serious as from the left tail.

10. Conclusions and Recommendations

The study has been carried out to investigate the performance of the Bayesian point and interval estimators of the parameters for Burr type VIII distribution based on five priors, three loss functions, and three censoring techniques. From the above analysis it can be concluded that amount of risk decreases and rate of convergence of estimates towards the true value increases with the increase in sample size. The performance of the inverse Levy prior has been the best among all priors under all cases. On the other hand, the quadratic loss function has been found to be the most efficient loss function under each prior and every censoring technique. Similarly, the estimates under singly type II censoring scheme are associated with the minimum amount of risks. So, on the whole, the use of inverse Levy prior along with quadratic loss function under singly type II censored samples can be preferred to estimate the parameter of Burr type VIII distribution.

Appendix

Derivation of Different Expressions/Formulae

The pdf of the Burr type VIII distribution given in (1) can be written as

The CDF of the Burr type VIII distribution used in (2) can be written as

The likelihood function presented in (8) can be expressed as

where

The generalized (joint) version of priors presented in (3)–(7) can be written as

Based on the density, kernel of the posterior distribution can be given as

The complete version of the posterior distribution can be presented as

The Bayes estimator under the LLF can be derived by using the following formula:

where 

Therefore,

The formula for the derivation of posterior risk under LLF is

where  

Therefore, the posterior risk under LLF can be given as

The formula for the derivation of the Bayes estimator under PLF is

where 

Therefore,

The formula for the derivation of posterior risk under PLF is

Therefore, the posterior risk under PLF is

The Bayes estimator under QLF can be derived by using the following formula:

where

Hence, the Bayes estimator under the QLF is

Similarly, the expression for the posterior risk under QLF is

The derivation of Bayes estimators and posterior risks under other censoring schemes can be obtained in a similar manner.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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