Journal of Quality and Reliability EngineeringVolume 2013 (2013), Article ID 896807, 6 pageshttp://dx.doi.org/10.1155/2013/896807

Research Article

Bayesian Estimation Based on Rayleigh Progressive Type II Censored Data with Binomial Removals

Reza Azimi¹ and Farhad Yaghmaei²

¹Department of Statistics, Parsabad Moghan Branch, Islamic Azad University, Parsabad Moghan, Iran²Faculty of Science, Golestan University, Gorgan, Golestan, Iran

Received 4 October 2012; Revised 27 January 2013; Accepted 3 February 2013

Academic Editor: Mohammad Modarres

Copyright © 2013 Reza Azimi and Farhad Yaghmaei. 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 study considers the estimation problem for the parameter and reliability function of Rayleigh distribution under progressive type II censoring with random removals, where the number of units removed at each failure time has a binomial distribution. We use the maximum likelihood and Bayesian procedures to obtain the estimators of parameter and reliability function of Rayleigh distribution. We also construct the confidence intervals for the parameter of Rayleigh distribution. Monte Carlo simulation method is used to generate a progressive type II censored data with binomial removals from Rayleigh distribution, and then these data are used to compute the point and interval estimations of the parameter and compare both the methods used with different random schemes.

1. Introduction

The Rayleigh distribution provides a population model which is useful in several areas of statistics. In the literature, many researcher studied properties of the Rayleigh distribution, particularly in life testing and reliability. Life testing experiments often deal with censored sample in order to estimate the parameters involved in the life distribution. The cumulative distribution function (CDF), probability density function (PDF), and the reliability function of the Rayleigh distribution with parameter are respectively. Inferences for in the Rayleigh distribution have been discussed by several authors. Harter and Moore [1] derived an explicit form for the maximum likelihood estimator (MLE) of based on type II censored data. Moreover, Bayesian estimation and prediction problems for the based on doubly censored sample have been considered by Fernández [2] and Raqab and Madi [3]. Wu et al. [4] have derived the Bayesian estimator and prediction intervals based on progressively type II censored samples. A recent account on progressive censoring schemes can be obtained in the monograph by Balakrishnan and Aggarwala [5] or in the excellent review article by Balakrishnan [6].

Suppose that units are placed on a life test and the experimenter decides beforehand quantity , the number of units, to be failed. Now at the time of the first failure, of the remaining surviving units are randomly removed from the experiment. Continuing, at the time of the second failure, of the remaining units are randomly drawn from the experiment. Finally, at the time of the th failure, all the remaining surviving units are removed from the experiment. Note that, in this scheme, are all prefixed. However, in some practical situations, these numbers may occur at random. For example, in some reliability experiments, an experimenter may decide that it is inappropriate or too dangerous to carry on the testing on some of the tested units even though these units have not failed. In such cases, the pattern of removal at each failure is random [7, 8]. This leads to progressive censoring with random removals (illustrated by Table 1). There have been several references about the statistical inference on lifetime distributions under progressive censoring with random removals. Wu et al. [9] Wu and Chang [10] considered the estimation problem for the Burr type-XII distribution and Pareto distribution based on progressively censored samples with random removals, where the number of units removed at each failure time has a discrete uniform removal pattern and binomial distribution.

Table 1: A schematic representation of the progressive type II censoring with binomial removals.

Process
The number in life testing
Failures
Binomial removals
Remains

1

1

2

1

⋯
⋯
⋯
⋯
⋯

1

1

0

2. Maximum Likelihood Estimation

Let be the ordered failure times out of randomly selected times, where is predetermined before the test. At the th failure, items are removed from the test. For progressive censoring with predetermined number of removals , the likelihood function can be defined as the following form: where . Equation (4) is derived conditional on . Each can be of any integer value between 0 and . It is different from progressive censoring with fixed removal that is a random number and is assumed to follow a binomial distribution with parameter . It means that each unit leaves with equal probability and the probability of units leaving after the th failure occurs is where . Furthermore, we assume that is independent of for all . The joint likelihood function of and can be found as where is the joint probability distribution of and in particular Therefore Substituting (1) and (2) into (6), the likelihood function takes the following form: Thus, the log-likelihood function takes the following form: The first partial derivatives of log-likelihood function with respect to and are Solving these equations in and , we get the MLE of and as in the following forms: It is clear that the MLE of reliability function, , with fixed , can be obtained by

3. Confidence Intervals for

Let . It is easy to show that is a progressively type II censored sample from the standard exponential distribution. For a fixed set of , let us consider the following transformation:

Thomas and Wilson [11] showed that the generalized spacings , as defined in (14), are all independent and identically distributed as standard exponential. Hence, has a Chi-square distribution with 2 degrees of freedom. We can also write the numerator of as the sum of -independent generalized spacings. That is, . Therefore, we can find that, conditionally on a fixed set of has a Chi-square distribution with degrees of freedom.

Confidence interval for can be obtained through the pivotal quantity and can be the percentile of Chi-square distribution with degrees of freedom and right-tail probability . Then since is increasing function of . Then confidence interval for is

4. Bayesian Estimation

In this section, we use the Bayes procedure to derive the point estimates of the parameters and and reliability function based on progressively type II censoring data with binomial removals. For this purpose we assume the parameters and behave as independent random variables. In this paper, for parameter we consider prior distribution of the form where and . This density is known as the square-root inverted-gamma distribution (see [12, page 119]) and, independently from parameter , has a beta prior distribution with parameters and and PDF Based on the prior and , the joint prior PDF of is It follows, from (9) and (20), that the joint posterior distribution of is given by where .

Therefore, the marginal posterior PDFs of and are given, respectively, by Substituting into (22), we obtain the posterior probability density function of as for .

4.1. Bayesian Estimation under Squared Error Loss Function

Under squared error loss, the Bayes estimator of is the mean of posterior density (22), which is given by And for parameter , the Bayes estimator of is Finally, the Bayes estimators of are found to be

4.2. Bayesian Estimation under Entropy Loss Function

In many practical situations, it appears to be more realistic to express the loss in terms of the ratio . In this case, Calabria and Pulcini [13] point out that a useful asymmetric loss function is the entropy loss function: where whose minimum occurs at . The latter has been used in Dey et al., [14] and Dey and Liu [15], in the original form having : Therefore, the Bayesian estimators of parameters under the entropy loss function with , denoted by , is given by the following equation: Therefore we have

4.3. Bayesian Estimation under Precautionary Loss Function

Precautionary loss function which is asymmetric, was introduced by Norström [16]: The Bayes estimator under the latter loss function is denoted by and may be obtained by solving the following equation: The Bayesian estimators of the parameters , and under asymmetric precautionary loss function are given, respectively, by

5. Numerical Study

In our study, we firstly generate the numbers of progressive censoring with binomial removals , and then we get the progressive censoring with binomial removals samples from Rayleigh distribution by the Monte-Carlo method. The steps are as follows.

Step 1. Generate a group values according to the relevant value of .

Step 2. Simulate -independent exponential random variables .This can be done using inverse transformation , where are independent uniform random variables.

Step 3. Set for . This is the required progressively type II censored sample with binomial removals from the standard exponential distribution.

Step 4. Finally, we set , for , where is the inverse cumulative distribution function of the Rayleigh distribution. Then is the required progressively type II censored sample with binomial removals from the Rayleigh distribution.

Step 5. We compute the MLE of parameters and by (12), for reliability function by (13) and confidence intervals by (16).

Step 6. We compute the the Bayes estimates of parameters and reliability function, respectively, using (25), (32), (37), (26), (33), (38), (27), (34), and (39).

Step 7. We repeat the above steps 2000 times. We then obtain the means and the MSEs (mean-squared error) for different censoring sizes and censoring schemes where and is the estimator of .

In all above cases the prior parameters chosen as ,   and which yield the generated value of as the true value. The true values of in is obtained . The results are summarized in Tables 2 and 3.

Table 2: Averaged values of MSEs for estimates of the parameters and .

MSE
MSE
MSE
MSE
CI
MSE
MSE
MSE
MSE

P = 0.1
10
5
0.9727
1.0484
1.0749
1.0019
(0.6796, 1.7070)
0.1333
0.2500
0.2105
0.1336

0.0514
0.0416
0.0469
0.0359
0.0011
0.0225
0.0122
0.0011

7
0.9794
1.0334
1.0520
0.9996
(0.7170, 1.5447)
0.0769
0.2222
0.1764
0.1139

0.0343
0.0292
0.0319
0.0263
0.0005
0.0149
0.0058
0.0001

20
10
0.9873
1.0250
1.0379
1.0009
(0.7554, 1.4258)
0.1228
0.1612
0.1475
0.0673

0.0251
0.0226
0.0239
0.0209
0.0005
0.0037
0.0022
0.0010

15
0.9889
1.0141
1.0226
0.9979
(0.7902, 1.3218)
0.0975
0.1521
0.1333
0.0627

0.0165
0.0153
0.0158
0.0146
5 × 10⁻⁶
0.0027
0.0011
0.0013

P = 0.4
10
5
0.9700
1.0461
1.0726
0.9997
(0.6778, 1.7024)
0.5000
0.5333
0.5000
0.4000

0.0522
0.0419
0.0471
0.0363
0.01000
0.0177
0.0100
0.0000

7
0.9805
1.0345
1.0532
1.0006
(0.7178, 1.5463)
0.2500
0.3846
0.3333
0.2517

0.0371
0.0318
0.0346
0.0287
0.0225
0.0002
0.0044
0.0219

20
10
0.9842
1.0221
1.0349
0.9980
(0.7530, 1.4213)
0.3703
0.4062
0.3870
0.2646

0.0243
0.0216
0.0229
0.0202
0.0008
0.0009
0.0001
0.0183

15
0.9897
1.0149
1.0234
0.9986
(0.7909, 1.3230)
0.4545
0.5000
0.4666
0.3638

0.0163
0.0151
0.0157
0.0144
0.0029
0.0100
0.0044
0.0013

P = 0.6
10
5
0.9706
1.0466
1.0730
1.0001
(0.6782, 1.7034)
0.7142
0.6666
0.6363
0.5547

0.0511
0.0411
0.0462
0.03557
0.0130
0.0044
0.0013
0.0020

7
0.9818
1.0356
1.0542
1.0017
(0.7188, 1.5485)
0.2727
0.3750
0.3333
0.2406

0.0343
0.0296
0.0323
0.0265
0.1071
0.0506
0.0711
0.1291

20
10
0.9906
1.0280
1.0409
1.0038
(0.7579, 1.4305)
0.6666
0.6500
0.6315
0.5307

0.0240
0.0219
0.0233
0.0201
0.0044
0.0025
0.0009
0.0047

15
0.9913
1.0164
1.0249
1.0001
(0.7922, 1.3251)
0.4545
0.5000
0.4666
0.3638

0.0161
0.0150
0.0156
0.0143
0.0211
0.0100
0.0177
0.0557

P = 0.9
10
5
0.9746
1.0500
1.0766
1.0034
(0.6810, 1.7104)
0.7142
0.6666
0.6363
0.5547

0.0499
0.0408
0.0461
0.0349
0.0344
0.0544
0.0695
0.1192

7
0.9779
1.0322
1.0508
0.9984
(0.7159, 1.5423)
0.7500
0.6666
0.6250
0.5577

0.0370
0.0315
0.0341
0.0285
0.0225
0.05447
0.0756
0.1171

20
10
0.9882
1.0258
1.0387
1.0017
(0.7560, 1.4270)
0.9090
0.8125
0.8000
0.7373

0.0248
0.0223
0.0237
0.0207
8 × 10⁻⁵
0.0076
0.0100
0.0264

15
0.9931
1.0181
1.0266
1.0018
(0.7936, 1.3274)
0.6250
0.6153
0.5833
0.4934

0.0153
0.0143
0.0149
0.0136
0.0756
0.0810
0.1002
0.1653

Table 3: Averaged values of MSEs for estimates of the reliability function.

MSE
MSE
MSE
MSE

P = 0.1
10
5
0.8578
0.8353
0.8297
0.8379

0.0052
0.0078
0.0094
0.0072

7
0.8662
0.8505
0.8472
0.8520

0.0030
0.0042
0.0048
0.0040

20
10
0.8714
0.8606
0.8586
0.8615

0.0018
0.0024
0.0026
0.0023

15
0.8764
0.8694
0.8682
0.8699

0.0010
0.0012
0.0013
0.0012

P = 0.4
10
5
0.8589
0.8366
0.8310
0.8392

0.0053
0.0079
0.0094
0.0072

7
0.8660
0.8503
0.8470
0.8519

0.0030
0.0042
0.0048
0.0040

20
10
0.8690
0.8581
0.8560
0.8590

0.0020
0.0026
0.0029
0.0025

15
0.8744
0.8673
0.8661
0.8679

0.0010
0.0013
0.0014
0.0012

P = 0.6
10
5
0.8592
0.8369
0.8313
0.8394

0.0052
0.0078
0.0092
0.0071

7
0.8672
0.8516
0.8484
0.8532

0.0030
0.0042
0.0047
0.0040

20
10
0.8710
0.8601
0.8582
0.8611

0.0018
0.0024
0.0026
0.0023

15
0.8749
0.8678
0.8666
0.8684

0.0010
0.0013
0.0014
0.0012

P = 0.9
10
5
0.8593
0.8370
0.8315
0.8396

0.0052
0.0077
0.0092
0.0071

7
0.8660
0.8503
0.8470
0.8518

0.0032
0.0045
0.0051
0.0043

20
10
0.8725
0.8617
0.8598
0.8626

0.0016
0.0022
0.0024
0.0021

15
0.8766
0.8695
0.8684
0.8700

0.0009
0.0011
0.0012
0.0011

6. Conclusions

We develop some results on Rayleigh distribution when progressive type II censoring with binomial removals is performed. Based on the results shown in Table 2, one can conclude that Bayesian estimators under precautionary loss function have the smallest biased and MSE as compared with the other estimators. The MSE associated with the estimators (Bayesian estimators under different loss functions and maximum likelihood estimators) of the parameter decrease with increasing the sample size . According to Table 3 maximum likelihood estimators of reliability function of Rayleigh distribution have smallest biased and MSE as compared with the other estimators. It is immediate to note that MSEs of reliability function estimators decrease as increases.

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