International Journal of Quality, Statistics, and ReliabilityVolume 2011 (2011), Article ID 618347, 10 pagesdoi:10.1155/2011/618347

Research Article

Bayes Estimation of a Two-Parameter Geometric Distribution under Multiply Type II Censoring

J. B. Shah¹ and M. N. Patel²

¹Department of Statistics, R. J. Tibrewal Commerce College, Vastrapur, Ahmedabad 380015, Gujarat, India²Department of Statistics, School of Sciences, Gujarat University, Ahmedabad, Gujarat 380009, India

Received 2 December 2010; Accepted 18 March 2011

Academic Editor: Kwai Sang Chin

Copyright © 2011 J. B. Shah and M. N. Patel. J. B. Shah and . 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.

We derive Bayes estimators of reliability and the parameters of a two- parameter geometric distribution under the general entropy loss, minimum expected loss and linex loss, functions for a noninformative as well as beta prior from multiply Type II censored data. We have studied the robustness of the estimators using simulation and we observed that the Bayes estimators of reliability and the parameters of a two-parameter geometric distribution under all the above loss functions appear to be robust with respect to the correct choice of the hyperparameters () and a wrong choice of the prior parameters () of the beta prior.

1. Introduction

In life testing experiments, a lot of work has been done under the continuous lifetime models. Sometimes it is neither possible nor convenient to measure the life length of an item continuously until its failure. Also failure time data is sometimes discrete either through the grouping of continuous data due to imprecise measurement or because time itself is discrete. In such circumstances one measures the life of a device on a discrete scale and considers the number of successful cycles, trials, or operations before failure. Therefore, the number of successful trials before failure is more pertinent than the time of continuous period. The one-parameter geometric distribution has an important position in discrete lifetime models. The geometric distribution can be used as a discrete failure to investigate the ability of electronic tubes to withstand successive voltage overloads and performance of electric switches, which are repeatedly turned on and off. Many authors like Yaqub and Khan [1], Patel and Gajjar [2], N. W. Patel and M. N. Patel [3] have contributed to the methodology and estimation of the parameter of the geometric distribution.

Here we assume that the lifetime of certain items has a two-parameter geometric distribution with probability mass function (pmf), cumulative distribution function (cdf), and reliability, respectively, as Here , and are unknown parameters.

This model is employed in situations where it is believed that death or failure cannot occur before certain cycles , where is the warranty time or threshold parameter and is the expected life of the item. One should be cautious when the model includes a threshold parameter , since the data usually does not provide enough information about and the inclusion of could cause rather special statistical problems. If the threshold parameter is known, then there is no difficulty. But if it is unknown then it must be estimated. The models with threshold parameter are difficult to fit with maximum likelihood because for such models likelihood may be unbounded and so maximum likelihood does not work.

In life testing experiments, the experimenter may not always be in a position to observe the lifetimes of all the items put on test. This may be because of time limitations and/or other restrictions on data collection. In this situations, Type I or Type II censoring schemes are used. Another way to get censored data is to censor first , middle and largest observations thus only available ordered observations are and . The scheme is known as the multiply Type II censoring scheme. This censoring scheme is useful in follow-up studies in epidemiology, reliability and endocrinology. Multiply Type II censoring has been considered by many authors like Balakrishnan [4], Fei and Kong [5], Kong and Fei [6] for continuous lifetime models.

In this paper, we have obtained Bayes estimators of parameters , and reliability of the life model defined in (1) when the data available are of the multiply Type II censored. Here we have considered three types of loss functions, namely, general entropy loss function (GELF), minimum expected loss function (MELF), and linex loss function (LLF). A comparison has been made using Bayes risk among the Bayes estimators. Using simulation study robustness of Bayes estimators has been studied.

2. Likelihood: Prior, Posterior, and Marginal

For discrete failure mode, the likelihood function under the multiply Type II censoring scheme described in previous section can be written as Using (1) and (2), the likelihood becomes proportional to where Here we consider beta prior density for of the form If the prior information about is given in terms of the prior mean and variances , respectively, then the prior parameters can be obtained by solving For , the above marginal prior reduces to diffuse (or natural) prior given by .

Also the Jeffreys’ invariant noninformative prior can be deduced from the above marginal prior with parameters and .

As in the stand by Broemeling and Tsurumi [7], we suppose the marginal prior distribution of to be discrete uniform, that is, Using (5) and (8), the joint prior distribution of and is The joint posterior density of and say , using (3) and (9), is where where .

Marginal posterior distributions of and are obtained as

3. Bayes Estimator

The Bayesian concepts and methods are found useful in Econometrics, Sociology, Engineering, Life Testing and Reliability estimation, Quality Control and others. The Bayes method avoids the possibility of an essentially non-negative parameter having a negative estimator which could occur when estimation is done using sampling framework. For the given sample and the corresponding likelihood function, Bayes estimator will be unique and there will be no confusion about the choice of the estimator. In estimating certain numbers of parameters in the presence of nuisance parameters (threshold parameters), there would be considerable difficulties in non-Bayesian case whereas in the Bayesian setup inference can be made very simple [8].

The Bayesian method usually requires less sample data to achieve the same quality of inference as compared to the method based on sampling theory. This is an especially important consideration in those areas of application where sample data may be either expensive or difficult to obtain, for example as in reliability. Over the years, reliability estimation methods based on sampling theory have been found to be extremely useful for a wide variety of problems. However, there are many instances in which the classical methods have been found to be less satisfactory. As reliability testing is directly related to cost and time, the use of sampling theory would be extremely limited. Sampling theory methods are especially inappropriate for reliability analysis based on scarce data.

Most of the engineering designs are evolutionary rather than revolutionary processes in which current equipment is modified to suit new requirements. In such instances it would appear that the known facts regarding reliability of the current hardware could possibly be utilized in an attempt to improve the quality of the reliability estimates of the new design. The sampling theory methods are inappropriate for incorporating such related information.

In reliability estimation, the Bayesian analysis has been found useful due to two facts: one is the increased quality of the inferences; provided the prior information accurately reflects the true variation in the parameter(s); the other is the reduction in testing requirements (like test time, sample size, etc.) that often occur in Bayesian reliability demonstration test programs. The Bayesian method produces believable results which go a long way in convincing the engineer of the utility of Bayesian methods.

4. Bayes Estimators under Minimum Expected Loss Function (MELF)

In Bayesian analysis, widely used loss function is a quadratic loss function given by If , it reduces to squared error loss function (SELF) and for , it becomes known as MELF introduced by Rao Tummala and Sathe [9] in their study.

The Bayes estimators of under MELF are given by For the threshold parameter , which is a positive integer quantity , the loss function is defined as , where = is the smallest integer greater than the analytical solution.

The Bayes estimators of and under MELF are

5. Bayes Estimators under General Entropy Loss Function (GELF)

Sometimes, the use of symmetric loss function, namely, SELF, was found to be inappropriate in a situation where an overestimation of the reliability function is usually much more serious than an underestimation. Here we consider the asymmetric loss function, namely, GELF, proposed by Calabria and Pulcini [10], is whereas for the threshold parameter , the loss function is defined as where , , and = . Here is the smallest integer greater than the analytical solution. The sign of the shape parameter reflects the deviation of the asymmetry, > 0 if overestimation is more serious than underestimation, and vice versa. The magnitude of reflects the degree of asymmetry. The Bayes estimator of under the GELF given in (17) is given by The Bayes estimators of , and under GELF are, respectively, The Bayes estimators of , and under SELF can be obtained by substituting in the results (20) to (22), respectively.

6. Bayes Estimators under Linex Loss Function (LLF)

Another asymmetric loss function given by Varian [11], known as LLF, is defined as whereas for the threshold parameter , the loss function is defined as where and = . Here is the smallest integer greater than the analytical solution. The sign and magnitude of the shape parameter reflect the deviation and degree of asymmetry, respectively. The Bayes estimator of under LLF given in (23) is given by Using marginal posterior distributions (12) in (25) and formula 3.478 (3), page 364 of Gradshteyn and Ryzhik [12], the Bayes estimators of , and , respectively, can be obtained as where is called a generalized hypergeometric series which is defined as follows (see Gradshteyn and Ryzhik (see [12, page 1000, formula 9.14 (1)]: where .

The underconsidered multiply Type II censoring scheme reduces to right Type II, left Type II, and doubly Type II censoring schemes when and , and and , respectively.

7. Numerical Example

We have generated 30 random observations with from the distribution defined in (1) having reliability at . Suppose that information about first 3, middle 3, and largest 3 observations are censored, then the only available observations are given in Table 1.

Table 1: Generated observations from geometric distributions.

I
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

—
—
—
3
3
3
3
3
3
3
3
3
3
4
4

i
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

4
4
4
—
—
—
6
6
6
6
6
6
—
—
—

In Table 1, . Prior parameters and themselves were random observations from beta distribution with means and variance , resulting in and . We have calculated Bayes estimators of and at = 6 under MELF, SELF, GELF, and LLF. For a comparative purpose, point estimates under the diffuse prior () and Jeffreys’ prior () have also been calculated.

The results are shown in Table 2.

Table 2: The values of bayes estimators of , and at .

Prior
Loss function
Shape parameter

MELF
—
2
0.3629
0.1184

GELF
−3
5
0.3868
0.1730

GELF
−2
5
0.3829
0.1643

SELF
−1
5
0.3790
0.1555

GELF
1
3
0.3711
0.1370

GELF
2
3
0.3669
0.1274

Informative
GELF
3
2
0.3628
0.1175

LLF
−3
7
0.3835
0.1590

LLF
−2
7
0.3820
0.1581

LLF
−1
6
0.3806
0.1568

LLF
1
3
0.3776
0.1541

LLF
2
2
0.3761
0.1531

LLF
3
2
0.3747
0.1530

Diffuse
MELF
—
2
0.3760
0.1028

GELF
−3
5
0.4020
0.1613

GELF
−2
5
0.3979
0.1520

SELF
−1
5
0.3937
0.1425

GELF
1
3
0.3850
0.1226

GELF
2
3
0.3805
0.1122

GELF
3
2
0.3759
0.1016

LLF
−3
7
0.3987
0.1461

LLF
−2
7
0.3970
0.1451

LLF
−1
6
0.3954
0.1439

LLF
1
3
0.3920
0.1411

LLF
2
2
0.3904
0.1401

LLF
3
2
0.3887
0.1397

Jeffreys’
MELF
—
2
0.3723
0.1043

GELF
−3
5
0.3992
0.1650

GELF
−2
5
0.3950
0.1554

SELF
−1
5
0.3906
0.1455

GELF
1
3
0.3816
0.1249

GELF
2
3
0.3769
0.1141

GELF
3
2
0.3722
0.1031

LLF
−3
7
0.4014
0.1440

LLF
−2
7
0.3996
0.1429

LLF
−1
6
0.3979
0.1417

LLF
1
3
0.3946
0.1389

LLF
2
2
0.3929
0.1379

LLF
3
2
0.3912
0.1374

From Table 2, we observed that the Bayes estimator of moves away from the true value of under for GELF and for LLF, but the Bayes estimators of the other parameters are not affected, so much, under these choices. So, in subsequent Sections 8 and 9, we have considered the values of only −2, −1, 1, and 2 whereas for only −2, −1, 1, and 2 for the comparison.

8. Sensitivity of Bayes Estimators

In this section, we have studied the sensitivity of the Bayes estimators with respect to changes in the parameters of the beta prior. We have computed Bayes estimators of and at in case of MELF, SELF, GELF, and LLF considering different sets of values of () for the data given in Table 1. The results are shown in Tables 3–6.

Table 3: The Bayes estimators and their risks under SELF and MELF.

a
b

SELF

MELF

1
5
0.3629
0.1720
2
0.3459
0.1325

2
5
0.3711
0.1634
2
0.3545
0.1252

7
3
5
0.3791
0.1555
2
0.3629
0.1184

5
5
0.3945
0.1409
2
0.3791
0.1061

7
5
0.4091
0.1279
2
0.3945
0.0954

9
5
0.4231
0.1165
2
0.4091
0.0860

1

5
0.4104
0.1277
2
0.3937
0.0909

3

5
0.3994
0.1370
2
0.3829
0.0999

5

5
0.3890
0.1462
2
0.3726
0.1092

7
3
5
0.3791
0.1555
2
0.3629
0.1184

8

5
0.3743
0.1600
2
0.3582
0.1231

9

5
0.3697
0.1646
2
0.3537
0.1277

Table 4: The Bayes estimators and their risks under GELF.

a
b

1
5
0.3670
0.1813
3
0.3545
0.1524
3
0.3501
0.1421

2
5
0.3751
0.1726
3
0.3629
0.1444
3
0.3587
0.1345

7
3
5
0.3830
0.1643
3
0.3711
0.1370
3
0.3670
0.1274

5
5
0.3982
0.1493
3
0.3869
0.1235
3
0.3830
0.1145

7
5
0.4126
0.1359
3
0.4019
0.1116
3
0.3982
0.1032

9
5
0.4265
0.1239
3
0.4162
0.1011
3
0.4126
0.0932

1

5
0.4144
0.1367
3
0.4022
0.1092
3
0.3979
0.0996

3

5
0.4033
0.1459
3
0.3912
0.1184
3
0.3870
0.1088

5

5
0.3929
0.1552
3
0.3809
0.1277
3
0.3767
0.1181

7
3
5
0.3830
0.1643
3
0.3711
0.1370
3
0.3670
0.1274

8

5
0.3782
0.1689
3
0.3664
0.1416
3
0.3623
0.1320

9

5
0.3735
0.1734
3
0.3618
0.1462
3
0.3577
0.1367

Table 5: The Bayes estimators and their risks under LLF.

a
b

1
6
0.3644
0.1736
7
0.3659
0.1749

2
6
0.3726
0.1650
7
0.3741
0.1662

7
3
6
0.3806
0.1568
7
0.3820
0.1581

5
6
0.3959
0.1421
7
0.3974
0.1431

7
6
0.4106
0.1290
7
0.4120
0.1299

9
6
0.4245
0.1174
7
0.4260
0.1182

1

6
0.4120
0.1289
7
0.4137
0.1300

3

6
0.4009
0.1382
7
0.4026
0.1394

5

6
0.3905
0.1476
7
0.3920
0.1487

7
3
6
0.3806
0.1568
7
0.3820
0.1581

8

6
0.3758
0.1615
7
0.3772
0.1627

9

6
0.3711
0.1661
7
0.3726
0.1673

Table 6: The Bayes estimators and their risks under LLF.

a
b

1
3
0.3614
0.1704
2
0.3599
0.1694

2
3
0.3696
0.1620
2
0.3681
0.1610

7
3
3
0.3776
0.1541
2
0.3761
0.1531

5
3
0.3930
0.1397
2
0.3916
0.1388

7
3
0.4072
0.1269
2
0.4062
0.1261

9
3
0.4217
0.1156
2
0.4203
0.1148

1

3
0.4088
0.1266
2
0.4071
0.1256

3

3
0.3978
0.1358
2
0.3962
0.1348

5

3
0.3874
0.1450
2
0.3859
0.1440

7
3
3
0.3776
0.1541
2
0.3761
0.1531

8

3
0.3729
0.1587
2
0.3714
0.1577

9

3
0.3683
0.1632
2
0.3669
0.1623

Following Calabria and Pulcini [13], we also assume the prior information to be correct if the true value of is close to prior mean and is assumed to be wrong if is far from .

We observed that the Bayes estimators of , and under MELF and SELF appear to be robust with respect to the correct choice of prior parameters a () and a wrong choice of the prior parameters (). This can be seen from Table 3. These conclusions are also valid in case of GELF and LLF, as observed in Tables 4, 5, and 6.

9. Simulation Results

In this section, we have generated 500 different random samples with from the distribution defined in (1) to justify the results obtained in Section 8 and obtained the Bayes estimators and their risk = under SELF, GELF, LLF and MELF for different choices of (). The results are summarized in Tables 7, 8, 9, and 10.

Table 7: The Bayes estimators and their risks under SELF and MELF.*

a
b

SELF

MELF

1
4
0.2961
0.2572
2
0.2810
0.2224

0.07469
0.01105
0.03011
0.11111
0.02336
0.07138

2
4
0.3034
0.2471
2
0.2887
0.2129

0.07411
0.01030
0.03207
0.11111
0.02140
0.07604

7
3
4
0.3106
0.2375
2
0.2961
0.2040

0.08128
0.01015
0.03591
0.11111
0.02073
0.08321

5
4
0.3244
0.2196
2
0.3106
0.1875

0.07528
0.01135
0.04931
0.11111
0.02290
0.10346

7
4
0.3377
0.2035
2
0.3244
0.1727

0.08362
0.01411
0.07050
0.11111
0.02929
0.12949

9
4
0.3505
0.1888
2
0.3377
0.1594

0.07996
0.01799
0.09969
0.11111
0.03938
0.15932

1

4
0.3316
0.2125
2
0.3167
0.1781

0.08098
0.01433
0.07098
0.11111
0.03006
0.13048

3

4
0.3243
0.2209
2
0.3095
0.1868

0.07981
0.01246
0.05556
0.11111
0.02558
0.11215

5

4
0.3173
0.2293
2
0.3027
0.1954

0.08245
0.01108
0.04414
0.11111
0.02253
0.09640

7
3
4
0.3106
0.2375
2
0.2961
0.2040

0.08128
0.01015
0.03591
0.11111
0.02073
0.08321

8

4
0.3073
0.2416
2
0.2929
0.2083

0.08347
0.00985
0.03279
0.11111
0.02024
0.07757

9

4
0.03042
0.2456
2
0.2898
0.2125

0.08259
0.00967
0.03026
0.11111
0.02000
0.07256

*In each cell the first value gives Bayes estimate and second its risk.

Table 8: The bayes estimators and their risks under GELF.*

a
b

1
5
0.2998
0.2654
3
0.2887
0.2400
3
0.2848
0.2310

0.38165
0.04352
0.11424
0.000
0.01142
0.03371
0.000
0.04648
0.14289

2
5
0.3070
0.2551
3
0.2961
0.2302
3
0.2924
0.2214

0.38165
0.04071
0.12074
0.000
0.01054
0.03619
0.000
0.04272
0.15354

7
3
5
0.3141
0.2454
3
0.3034
0.2209
3
0.2998
0.2123

0.38165
0.04029
0.13513
0.000
0.01031
0.04029
0.000
0.04154
0.17015

5
5
0.3278
0.2273
3
0.3176
0.2037
3
0.3141
0.1955

0.38165
0.04513
0.18951
0.000
0.01147
0.05291
0.000
0.04608
0.21865

7
5
0.3409
0.2108
3
0.3312
0.1882
3
0.3278
0.1803

0.38165
0.05565
0.28154
0.000
0.01445
0.07071
0.000
0.05858
0.28386

9
5
0.3536
0.1959
3
0.3442
0.1742
3
0.3410
0.1666

0.38165
0.07015
0.41591
0.000
0.01895
0.09296
0.000
0.07779
0.36221

1

5
0.3352
0.2207
3
0.3242
0.1954
3
0.3204
0.1865

0.38165
0.05614
0.28525
0.000
0.01477
0.07138
0.000
0.05999
0.28830

3

5
0.3279
0.2290
3
0.3170
0.2040
3
0.3132
0.1952

0.38165
0.04933
0.21658
0.000
0.01271
0.05856
0.000
0.05130
0.24075

5

5
0.3208
0.2373
3
0.3101
0.2125
3
0.3063
0.2038

0.38165
0.04397
0.16824
0.000
0.01124
0.04829
0.000
0.04524
0.20161

7
3
5
0.3141
0.2454
3
0.3034
0.2209
3
0.2998
0.2123

0.38165
0.04029
0.13513
0.000
0.01031
0.04029
0.000
0.04154
0.17015

8

5
0.3108
0.2494
3
0.3002
0.2251
3
0.2966
0.2165

0.38165
0.03907
0.12310
0.000
0.01003
0.03706
0.000
0.04049
0.15711

9

5
0.3076
0.2534
3
0.2971
0.2293
3
0.2935
0.2207

0.38165
0.03825
0.11361
0.000
0.00987
0.03429
0.000
0.03994
0.14575

*In each cell the first value gives Bayes estimate and second its risk.

Table 9: The bayes estimators and their risks under LLF.*

a
b

1
6
0.2972
0.2591
7
0.2983
0.2598

2.0498
0.00094
0.00185
7.0003
0.00377
0.00717

2
6
0.3045
0.2489
7
0.3056
0.2497

2.0498
0.00096
0.00167
7.0003
0.00387
0.00653

7
3
6
0.3117
0.2392
7
0.3128
0.2400

2.0498
0.00104
0.00016
7.0003
0.00417
0.00628

5
6
0.3255
0.2212
7
0.3267
0.2221

2.0498
0.00132
0.00170
7.0003
0.00530
0.00675

7
6
0.3389
0.2049
7
0.3400
0.2058

2.0498
0.00177
0.00206
7.0003
0.00706
0.00828

9
6
0.3516
0.1901
7
0.3527
0.1909

2.0498
0.00236
0.00262
7.0003
0.00935
0.01061

1

6
0.3329
0.2142
7
0.3341
0.2151

2.0498
0.00177
0.00210
7.0003
0.00705
0.00842

3

6
0.3255
0.2226
7
0.3267
0.2235

2.0498
0.00146
0.00186
7.0003
0.00584
0.00741

5

6
0.3184
0.2309
7
0.3196
0.2318

2.0498
0.00122
0.00170
7.0003
0.00489
0.00670

7
3
6
0.3117
0.2392
7
0.3128
0.2400

2.0498
0.00104
0.00016
7.0003
0.00417
0.00628

8

6
0.3084
0.2432
7
0.3095
0.2441

2.0498
0.00097
0.00157
7.0003
0.00389
0.00617

9

6
0.3052
0.2473
7
0.3063
0.2481

2.0498
0.00091
0.00156
7.0003
0.00366
0.00612

*In each cell the first value gives Bayes estimate and second its risk.

Table 10: The bayes estimators and their risks under LLF.

a
b

1
3
0.2950
0.2555
2
0.2940
0.2572

0.000
0.00093
0.00186
1.1353
0.00369
0.00845

2
3
0.3023
0.2455
2
0.3013
0.2467

0.000
0.00095
0.00168
1.1353
0.00378
0.00742

7
3
3
0.3095
0.2359
2
0.3084
0.2368

0.000
0.00102
0.00160
1.1353
0.00407
0.00690

5
3
0.3233
0.2182
2
0.3223
0.2168

0.000
0.00131
0.00169
1.1353
0.00524
0.00698

7
3
0.3367
0.2022
2
0.3356
0.2022

0.000
0.00178
0.00204
1.1353
0.00711
0.00817

9
3
0.3494
0.1876
2
0.3483
0.1874

0.000
0.00239
0.00256
1.1353
0.00963
0.01014

1

3
0.3304
0.2110
2
0.3292
0.2112

0.000
0.00177
0.00208
1.1353
0.00711
0.00850

3

3
0.3231
0.2194
2
0.3220
0.2198

0.000
0.00145
0.00185
1.1353
0.00580
0.00769

5

3
0.3162
0.2277
2
0.3150
0.2283

0.000
0.00121
0.00169
1.1353
0.00480
0.00716

7
3
3
0.3095
0.2359
2
0.3084
0.2368

0.000
0.00102
0.00160
1.1353
0.00407
0.00690

8

3
0.3063
0.2400
2
0.3052
0.2410

0.000
0.00096
0.00158
1.1353
0.00380
0.00688

9

3
0.3031
0.2440
2
0.2950
0.2553

0.000
0.00090
0.00157
1.1353
0.00347
0.00780

*In each cell the first value gives Bayes estimate and second its risk.

10. Conclusions

From Table 7, we observed that the Bayes estimators of the parameter perform well (minimum Bayes risk) under MELF compared to SELF for any choice of hyper parameters and .

From Tables 8–10, we find that under both the asymmetric loss functions, namely, GELF and LLF, Bayes risks become smaller when parameters ( and ) tend to zero, that is, Bayes estimators perform well under these asymmetric loss functions.

From Tables 7–10 following Calabria and Pulcini [13], we also observed that the Bayes estimators of the parameters are robust with respect to the correct choice of prior parameters and a wrong choice of the prior parameters which we have concluded in Section 8.

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