ISRN Probability and Statistics

Volume 2012 (2012), Article ID 292384, 15 pages

doi:10.5402/2012/292384

Research Article

Inference for the Geometric Extreme Exponential Distribution under Progressive Type II Censoring

Reza Pakyari

Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran

Received 22 June 2012; Accepted 11 July 2012

Academic Editors: M. Galea and M. Montero

Copyright © 2012 Reza Pakyari. 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.

Geometric extreme exponential (GE-exponential) is one of the nonnegative right-skewed distribution that is suitable for analyzing lifetime data. It is well known that the maximum likelihood estimators (MLEs) of the parameters lead to likelihood equations that have to be solved numerically. In this paper, we provide explicit estimators through an approximation of the likelihood equations based on progressively Type-II-censored samples. The approximate estimators are then used as starting values to find the MLEs numerically. The bias and variances of the MLEs are calculated for a wide range of sample sizes and different progressive censoring schemes through a Monte Carlo simulation study. Moreover, formulas for the observed Fisher information are given which could be used to construct asymptotic confidence intervals. The coverage probabilities of the confidence intervals and the percentage points of pivotal quantities associated with the MLEs are also calculated. A real dataset has been studied for illustrative purposes.

1. Introduction

Progressive censoring is one of the important sampling techniques that was first introduced by Herd [1] and its importance in life testing reliability experiments was discussed by Cohen [2].

The progressive Type-II censoring is as follows. Suppose units are placed on test. At the time of the first failure, units are randomly removed from the surviving units. Next, at the time of the second failure, units are randomly removed from the surviving units, and so on. Finally, after the th failure, all remaining units are removed. Thus, we observe complete failures and items are progressively censored from the units under test, and so . The vector is called the censoring scheme and is fixed prior to the study. If , no withdrawals are made which correspond to the complete sample and the ordinary order statistics will be obtained. If , we obtain the conventional Type-II right censoring. We will denote the progressive Type-II censored data by . For an excellent discussion on progressive Type-II censoring technique see the monograph of Balakrishnan and Aggarwala [3] and the recent discussion paper by Balakrishnan [4].

There are numerous articles in the literature dealing with inferential procedures based on the progressively Type-II censoring data for a wide variety of lifetime distributions. See for example Mann [5, 6], Balakrishnan et al. [7], Balakrishnan and Kannan [8], Balakrishnan et al. [9], Kim and Han [10], Lin et al. [11], and Asgharzadeh [12] among others who study the inferential procedures for the Weibull, extreme value, logistic, normal, half-logistic, log-gamma, and generalized exponential models, respectively.

GE-exponential distribution was introduced by Marshal and Olkin [13] and further studied extensively by Adamidis and Loukas [14] and Adamidis et al. [15]. Recently, this model was of interest of authors as a natural competitor to the other well-established lifetime models such as Weibull, lognormal, and gamma distributions (see, e.g. [16, 17]).

In this paper, we study the inference for the parameters of the GE-exponential distribution based on progressively Type-II censored data.

The rest of the paper is organized as follows. The MLEs of the parameters are studied in Section 2. In Section 3, we provide an approximation to the likelihood function which leads to explicit approximate estimators of the parameters. The observed Fisher information is studied in Section 4. The performance of the maximum likelihood estimators is evaluated in terms of bias and variance through a large number of Monte Carlo simulations and the results are given in Section 5. Using the asymptotic normality of the proposed estimators, the coverage probabilities of the asymptotic confidence intervals and the percentage points for the pivotal quantities are calculated by simulations and the results are provided in Section 6. Finally, Section 7 is devoted to study a real dataset for illustration of the proposed methodology.

2. Maximum Likelihood Estimation

Let the failure times follow a geometric extreme exponential distribution with shape parameter and scale parameter and with probability density function:

where . The cumulative distribution function is give by

The GE-exponential reduces to the exponential distribution if we take . When , the distribution reduces to exponential geometric distribution studied by Adamidis and Loukas [14]. The GE-exponential distribution has decreasing failure rate when and increasing failure rate when . The density function is unimodal with modal value equal to when . Hereafter, we assume .

Let be a progressively Type-II censored sample from density function (2.1). The likelihood function based on the censored sample is given by (see [3])

where

The log-likelihood function may be written as

where is a constant. Let , now the log-likelihood function may be rewritten as

The likelihood equations are then given by

which may be simplified as

The MLEs of the parameters and are the solutions of the system of nonlinear (2.9) and (2.10) which cannot be expressed explicitly. Standard numerical procedures such as Newton’s method needed to solve the above system of nonlinear equations. However, as is necessary for every numerical method, are starting values are needed to implement the numerical procedure. In the following section, we present an approximate MLEs for the parameters and to be used as starting values for implementation of the Newton’s method to solve the system of nonlinear (2.9) and (2.10). The approximation method is based on first finding a moment estimator for the shape parameter and based on this estimator finding an approximate MLE of the scale parameter and using that to find an approximate MLE of by either (2.9) or (2.10).

3. Approximate Maximum Likelihood Estimators

As was studied in the previous section, the maximum likelihood estimators of the parameters are given only implicitly by solving the system of nonlinear equations (2.9) and (2.10) by an appropriate numerical method. In this section, we provide approximate maximum likelihood estimators to be used as starting values to solve the system of nonlinear equations numerically. We propose a two-step approximation method, in which the moment estimator of the shape parameter is used to find the approximate maximum likelihood estimator of the scale parameter . The approximate MLE of is then can be found by either the likelihood equations (2.9) or (2.10) and the value of the approximate MLE of .

To find a moment estimator of , we consider the coefficient of variation (C.V.) which is given by

where is the Lerch’s transcendent (see, for example, [18]).

Note that C.V. is independent from the scale parameter and depends only on the shape parameter . Table 1 provides the value of the C.V. for different values of the shape parameter . This table may be used to find the moment estimator of the shape parameter for the given value of the sample C.V. when the shape parameter is unknown.

Table 1: The values of the coefficient of variation (C.V.) for different values of the shape parameter for the GE-exponential distribution.

C.V.

C.V.

C.V.

C.V.

0.1
1.847
2.2
0.825
6.5
0.642
17.0
0.522

0.2
1.523
2.4
0.808
7.0
0.631
18.0
0.516

0.3
1.365
2.6
0.792
7.5
0.622
19.0
0.510

0.4
1.265
2.8
0.779
8.0
0.613
20.0
0.505

0.5
1.193
3.0
0.766
8.5
0.605
21.0
0.500

0.6
1.138
3.2
0.755
9.0
0.598
22.0
0.495

0.7
1.094
3.4
0.744
9.5
0.591
23.0
0.491

0.8
1.058
3.6
0.734
10.0
0.584
24.0
0.487

0.9
1.027
3.8
0.725
10.5
0.578
25.0
0.483

1.0
1.000
4.0
0.717
11.0
0.572
26.0
0.479

1.1
0.976
4.2
0.709
11.5
0.567
27.0
0.475

1.2
0.956
4.4
0.701
12.0
0.562
28.0
0.472

1.3
0.937
4.6
0.694
12.5
0.557
29.0
0.468

1.4
0.920
4.8
0.687
13.0
0.552
30.0
0.465

1.5
0.905
5.0
0.681
13.5
0.548
35.0
0.452

1.6
0.890
5.2
0.675
14.0
0.544
40.0
0.440

1.7
0.877
5.4
0.669
14.5
0.540
45.0
0.430

1.8
0.865
5.6
0.664
15.0
0.536
50.0
0.422

1.9
1.854
5.8
0.659
15.5
0.532
55.0
0.414

2.0
1.844
6.0
0.654
16.0
0.529
60.0
0.408

Let be a progressively Type-II censored sample from , where is the moment estimator of . The likelihood equation (2.9) may now be rewritten in terms of as

where . The MLE of the scale parameter is the solution of the nonlinear equation (3.2) which can not be expressed explicitly.

Let us approximate the likelihood equation (3.2) by approximating the function in a Taylor series expansion around . From Balakrishnan and Sandhu [19] we have

where is the th order statistic of a progressively Type-II censored sample from the U distribution.

We then have

and hence

where and

Expanding around by Taylor series expansion and keeping the first two terms, we deduce

where

Therefore, the likelihood equation (3.2) may be approximated by

Equation (3.9) may be rewritten as

or

where

Equation (3.11) is a quadratic equation in terms of with the following two solutions:

However, one of the solutions is inadmissible since . Hence, the approximate MLE of the scale parameter is given explicitly by

The approximate MLE of , say is then maybe found by plugging in the value of in either (2.9) or (2.10) and solve the equation by a numerical method. Once the values of and are found, we could use them as good starting values to solve the likelihood (2.9) and (2.10) numerically in order to find the MLEs of and .

4. Fisher Information

In this section, we derive the asymptotic variance-covariance matrix of the maximum likelihood estimators of the parameters and .

The observed Fisher information may be obtained from the likelihood equations (2.7) and (2.8) as

where

The asymptotic variance-covariance matrix of the MLEs is then given by inverting the observed Fisher information matrix:

where

These quantities are estimated by evaluating them at MLEs

where and are the MLEs of and , respectively. We will use these asymptotic variances on Section 6 to derive asymptotic confidence intervals for the scale parameter and the shape parameter .

5. Simulation Study

In this section, we assess the performance of the proposed maximum likelihood estimators by a large number of Monte Carlo simulation experiments. All the simulations were carried out in R using the pseudorandom generator in that software package. We generated the progressively Type-II censored samples from the GE-exponential distribution with and using the algorithm presented by Balakrishnan and Sadhu [19]. The choice of provides unimodality and increasing failure rate.

Five values of the sample sizes, say , different choices of the effective sample size , and different progressive schemes were considered. For brevity in notation, we follow the Balakrishnan, et al. [7] notation to denote the censoring scheme. For example, denotes the censoring scheme .

We first calculated the moment estimator of the shape parameter using sample C.V. and Table 1. Then the approximate MLE of the scale parameter was calculated using (3.14). Approximate MLE of was then calculated by solving (2.8) numerically. The MLEs of and were calculated by solving the system of normal equations (2.7) and (2.8) simultaneously using BB R package of Varadhan and Gilbert [20]. We repeated this procedure 10000 times.

Tables 2 and 3 give the average of absolute bias, variance, and estimated variance calculated from observed Fisher information for MLEs and , respectively. The values of the absolute biases and variances are reasonably small for almost all the sample sizes and censoring schemes indicating the validity of the MLEs and . Also, as one would expect, both the bias and the variance decrease as the effective sample proportion that is, increases. Moreover, the observed Fisher information tends to be close to the variance of the estimator for both the MLEs when the effective sample proportion increases.

Table 2: Absolute bias, variance, and variance from observed Fisher information for the MLE of .

Scheme
AB()
var()

15
5
(10, 4 * 0)
0.0845
0.1124
0.1087

15
5
(4 * 0, 10)
0.0876
0.1174
0.1113

15
5
(5 * 2)
0.0862
0.1156
0.1145

15
6
(9, 5 * 0)
0.0745
0.0963
0.0917

15
6
(5 * 0, 9)
0.0810
0.1035
0.0943

15
6
(3 * 3, 3 * 0)
0.0788
0.1012
0.0922

15
7
(8, 6 * 0)
0.0689
0.0897
0.0842

15
7
(6 * 0, 8)
0.0736
0.0940
0.0912

15
7
(3 * 0, 4 * 2)
0.0755
0.0928
0.0892

20
6
(14, 5 * 0)
0.0644
0.0912
0.0878

20
6
(5 * 0, 14)
0.0689
0.0964
0.0912

20
6
(2 * 7, 4 * 0)
0.0674
0.0936
0.0890

20
8
(12, 7 * 0)
0.0574
0.0734
0.0701

20
8
(7 * 0, 12)
0.0619
0.0792
0.0723

20
8
(3 * 4, 5 * 0)
0.0604
0.0756
0.0716

20
10
(10, 9 * 0)
0.0511
0.0612
0.0579

20
10
(9 * 0, 10)
0.0556
0.0641
0.0618

20
10
(10 * 1)
0.0544
0.0632
0.0609

30
8
(22, 7 * 0)
0.0513
0.0667
0.0634

30
8
(7 * 0, 22)
0.0576
0.0689
0.0655

30
8
(8, 8, 6, 5 * 0)
0.0566
0.0671
0.0640

30
10
(20, 9 * 0)
0.0409
0.0581
0.0487

30
10
(9 * 0, 20)
0.0452
0.0621
0.0595

30
10
(10 * 2)
0.0444
0.0604
0.0543

30
15
(15, 14 * 0)
0.0346
0.0497
0.0442

30
15
(14 * 0, 15)
0.0380
0.0544
0.0507

30
15
(15 * 1)
0.0354
0.0530
0.0487

40
15
(25, 14 * 0)
0.0276
0.0482
0.0424

40
15
(14 * 0, 25)
0.0312
0.0512
0.0464

40
15
(10, 8, 7, 12 * 0)
0.0283
0.0491
0.0448

40
20
(20, 19 * 0)
0.0214
0.0423
0.0394

40
20
(19 * 0, 20)
0.0288
0.0449
0.0412

40
20
(20 * 1)
0.0294
0.0433
0.0407

50
20
(30, 19 * 0)
0.0204
0.0387
0.0325

50
20
(19 * 0, 30)
0.0298
0.0422
0.0401

50
20
(3 * 10, 17 * 0)
0.0254
0.0408
0.0385

50
25
(25, 24 * 0)
0.0140
0.0319
0.0279

50
25
(24 * 0, 25)
0.0199
0.0335
0.0308

50
25
(25 * 1)
0.0194
0.0324
0.0309

Table 3: Absolute bias, variance, and variance from observed Fisher information for the MLE of .

Scheme
AB()
var()

15
5
(10, 4 * 0)
0.1145
0.1328
0.1287

15
5
(4 * 0, 10)
0.1222
0.1354
0.1279

15
5
(5 * 2)
0.1182
0.1336
0.1259

15
6
(9, 5 * 0)
0.0954
0.1186
0.1078

15
6
(5 * 0, 9)
0.1024
0.1245
0.1165

15
6
(3 * 3, 3 * 0)
0.1011
0.1200
0.1099

15
7
(8, 6 * 0)
0.0815
0.0969
0.0888

15
7
(6 * 0, 8)
0.0878
0.1024
0.0992

15
7
(3 * 0, 4 * 2)
0.0855
0.0988
0.0892

20
6
(14, 5 * 0)
0.0834
0.1012
0.0985

20
6
(5 * 0, 14)
0.0889
0.1045
0.1028

20
6
(2 * 7, 4 * 0)
0.0874
0.1022
0.1001

20
8
(12, 7 * 0)
0.0712
0.0834
0.0806

20
8
(7 * 0, 12)
0.0769
0.0912
0.0856

20
8
(3 * 4, 5 * 0)
0.0750
0.0878
0.0817

20
10
(10, 9 * 0)
0.0742
0.0812
0.0785

20
10
(9 * 0, 10)
0.0785
0.0873
0.0814

20
10
(10 * 1)
0.0766
0.0832
0.0804

30
8
(22, 7 * 0)
0.0643
0.0912
0.0825

30
8
(7 * 0, 22)
0.0692
0.0945
0.0854

30
8
(8, 8, 6, 5 * 0)
0.0664
0.0842
0.0833

30
10
(20, 9 * 0)
0.0546
0.0722
0.0689

30
10
(9 * 0, 20)
0.0612
0.0780
0.0729

30
10
(10 * 2)
0.0594
0.0755
0.0680

30
15
(15, 14 * 0)
0.0392
0.0571
0.0536

30
15
(14 * 0, 15)
0.0472
0.0633
0.0582

30
15
(15 * 1)
0.0469
0.0627
0.0573

40
15
(25, 14 * 0)
0.0326
0.0574
0.0489

40
15
(14 * 0, 25)
0.0385
0.0595
0.0536

40
15
(10, 8, 7, 12 * 0)
0.0342
0.0570
0.0569

40
20
(20, 19 * 0)
0.0214
0.0423
0.0394

40
20
(19 * 0, 20)
0.0288
0.0449
0.0412

40
20
(20 * 1)
0.0294
0.0433
0.0407

50
20
(30, 19 * 0)
0.0279
0.0457
0.0398

50
20
(19 * 0, 30)
0.0356
0.0482
0.0441

50
20
(3 * 10, 17 * 0)
0.0254
0.0408
0.0385

50
25
(25, 24 * 0)
0.0289
0.0387
0.0356

50
25
(24 * 0, 25)
0.0260
0.0415
0.0388

50
25
(25 * 1)
0.0289
0.0393
0.0370

The tables also show that for all sample sizes and effective sample sizes , the censoring scheme tends to the smallest bias and variance, whilst the censoring scheme tends to the largest results. Incidentally, this is in coincidence with the findings of Balakrishnan et al. [9] for Gaussian distribution.

6. Asymptotic Confidence Intervals

In this section, we study the asymptotic confidence intervals for the parameters and using the estimation for the observed Fisher information matrix. Note that the asymptotic distributions of the pivotal quantities

are standard normal distribution. The asymptotic confidence intervals for the parameters are then given by

where is the upper critical value of the standard normal distribution.

Table 4 provides the coverage probabilities through 10000 Monte Carlo simulations for the parameters and . It is observed that the coverage probabilities are very close to their nominated values even for small sample sizes for both confidence intervals.

Table 4: 90% (First column) and 95% (Second column) coverage probabilities for the parameters and .

Scheme

15
5
(10, 4 * 0)
0.9089
0.9406
0.8612
0.9035

15
5
(4 * 0, 10)
0.9074
0.9523
0.8642
0.9042

15
5
(5 * 2)
0.9027
0.9504
0.8781
0.9089

15
6
(9, 5 * 0)
0.9086
0.9512
0.8697
0.9125

15
6
(5 * 0, 9)
0.9108
0.9508
0.8723
0.9154

15
6
(3 * 3, 3 * 0)
0.9084
0.9521
0.8706
0.9123

15
7
(8, 6 * 0)
0.9145
0.9571
0.8785
0.9241

15
7
(6 * 0, 8)
0.9092
0.9574
0.8717
0.9224

15
7
(3 * 0, 4 * 2)
0.9071
0.9514
0.8792
0.9235

20
6
(14, 5 * 0)
0.9136
0.9571
0.8714
0.9106

20
6
(5 * 0, 14)
0.9074
0.9583
0.8628
0.9164

20
6
(2 * 7, 4 * 0)
0.9047
0.9538
0.8705
0.9188

20
8
(12, 7 * 0)
0.9068
0.9579
0.8752
0.9178

20
8
(7 * 0, 12)
0.9008
0.9479
0.8666
0.9232

20
8
(3 * 4, 5 * 0)
0.9064
0.9587
0.8661
0.9213

20
10
(10, 9 * 0)
0.9034
0.9513
0.8886
0.9274

20
10
(9 * 0, 10)
0.9063
0.9517
0.8708
0.9341

20
10
(10 * 1)
0.9025
0.9516
0.8841
0.9378

30
8
(22, 7 * 0)
0.9187
0.9574
0.8706
0.9047

30
8
(7 * 0, 22)
0.9023
0.9508
0.8678
0.9314

30
8
(8, 8, 6, 5 * 0)
0.9014
0.9614
0.8617
0.9264

30
10
(20, 9 * 0)
0.9035
0.9531
0.8756
0.9218

30
10
(9 * 0, 20)
0.9098
0.9618
0.8707
0.9246

30
10
(10 * 2)
0.9059
0.9503
0.8713
0.9279

30
15
(15, 14 * 0)
0.9081
0.9530
0.8966
0.9308

30
15
(14 * 0, 15)
0.9146
0.9618
0.8942
0.9446

30
15
(15 * 1)
0.9042
0.9589
0.8914
0.9446

40
15
(25, 14 * 0)
0.9184
0.9564
0.8813
0.9324

40
15
(14 * 0, 25)
0.9016
0.9508
0.8863
0.9474

40
15
(10, 8, 7, 12 * 0)
0.9023
0.9574
0.8848
0.9358

40
20
(20, 19 * 0)
0.9065
0.9574
0.8965
0.9345

40
20
(19 * 0, 20)
0.9082
0.9549
0.8823
0.9349

40
20
(20 * 1)
0.9028
0.9526
0.8754
0.9377

50
20
(30, 19 * 0)
0.9048
0.9508
0.8685
0.9305

50
20
(19 * 0, 30)
0.9092
0.9534
0.8872
0.9375

50
20
(3 * 10, 17 * 0)
0.9047
0.9549
0.8863
0.9347

50
25
(25, 24 * 0)
0.9145
0.9541
0.8968
0.9417

50
25
(24 * 0, 25)
0.9095
0.9549
0.8927
0.9429

50
25
(25 * 1)
0.9080
0.9564
0.8933
0.9448

We also calculated the unconditional percentage points of these pivotal quantities and the results are given in Table 5. The values of the percentage points associate with and support the standard normality of these pivotal quantities specifically for large values of the sample size and the effective sample proportion. Indeed, the percentage points in Table 5 could be used to construct another asymptotic confidence intervals for the unknown parameters.

Table 5: (5.0, 95.0) (First column) and (2.5, 97.5) (Second column) percentage points for the pivotal quantities.

Scheme

15
5
(10, 4 * 0)
(−1.54, 1.61)
(−1.94, 1.85)
(−1.08, 2.65)
(−1.24, 3.39)

15
5
(4 * 0, 10)
(−1.73, 1.52)
(−2.18, 1.75)
(−1.64, 2.12)
(−2.00, 2.63)

15
5
(5 * 2)
(−1.59, 1.62)
(−2.00, 1.85)
(−1.22, 2.48)
(−1.44, 3.13)

15
6
(9, 5 * 0)
(−1.56, 1.63)
(−1.94, 1.89)
(−1.18, 2.44)
(−1.38, 3.08)

15
6
(5 * 0, 9)
(−1.70, 1.54)
(−2.15, 1.77)
(−1.64, 2.04)
(−1.99, 2.53)

15
6
(3 * 3, 3 * 0)
(−1.68, 1.55)
(−2.11, 1.78)
(−1.58, 2.12)
(−1.91, 2.61)

15
7
(8, 6 * 0)
(−1.58, 1.63)
(−1.93, 1.88)
(−1.26, 2.77)
(−1.48, 2.82)

15
7
(6 * 0, 8)
(−1.72, 1.52)
(−2.12, 1.77)
(−1.66, 1.94)
(−1.99, 2.42)

15
7
(3 * 0, 4 * 2)
(−1.61, 1.62)
(−1.97, 1.87)
(−1.32, 2.22)
(−1.55, 2.75)

20
6
(14, 5 * 0)
(−1.55, 1.61)
(−1.93, 1.85)
(−1.07, 2.59)
(−1.21, 3.27)

20
6
(5 * 0, 14)
(−1.70, 1.54)
(−2.15, 1.77)
(−1.62, 2.05)
(−1.98, 2.54)

20
6
(2 * 7, 4 * 0)
(−1.70, 1.54)
(−2.14, 1.76)
(−1.59, 2.10)
(−1.90, 2.59)

20
8
(12, 7 * 0)
(−1.57, 1.64)
(−1.93, 1.88)
(−1.23, 2.31)
(−1.44, 2.80)

20
8
(7 * 0, 12)
(−1.73, 1.55)
(−2.12, 1.78)
(−1.64, 1.95)
(−1.97, 2.36)

20
8
(3 * 4, 5 * 0)
(−1.71, 1.56)
(−2.10, 1.78)
(−1.60, 1.99)
(−1.92, 2.42)

20
10
(10, 9 * 0)
(−1.59, 1.65)
(−1.96, 1.91)
(−1.33, 2.15)
(−1.58, 2.60)

20
10
(9 * 0, 10)
(−1.70, 1.56)
(−2.12, 1.81)
(−1.64, 1.90)
(−1.97, 2.31)

20
10
(10 * 1)
(−1.63, 1.62)
(−2.01, 1.87)
(−1.45, 2.04)
(−1.72, 2.48)

30
8
(22, 7 * 0)
(−1.56, 1.62)
(−1.93, 1.83)
(−1.08, 2.48)
(−1.22, 3.01)

30
8
(7 * 0, 22)
(−1.73, 1.55)
(−2.12, 1.78)
(−1.64, 1.96)
(−1.96, 2.38)

30
8
(8, 8, 6, 5 * 0)
(−1.71, 1.55)
(−2.11, 1.77)
(−1.57, 2.03)
(−1.89, 2.47)

30
10
(20, 9 * 0)
(−1.57, 1.64)
(−1.93, 1.89)
(−1.20, 2.30)
(−1.40, 2.81)

30
10
(9 * 0, 20)
(−1.71, 1.56)
(−2.13, 1.81)
(−1.64, 1.91)
(−1.97, 2.32)

30
10
(10 * 2)
(−1.61, 1.64)
(−1.97, 1.88)
(−1.32, 2.17)
(−1.56, 2.63)

30
15
(15, 14 * 0)
(−1.57, 1.64)
(−1.89, 1.92)
(−1.35, 2.01)
(−1.58, 2.44)

30
15
(14 * 0, 15)
(−1.68, 1.56)
(−1.99, 1.84)
(−1.62, 1.80)
(−1.89, 2.19)

30
15
(15 * 1)
(−1.62, 1.61)
(−1.95, 1.87)
(−1.46, 1.93)
(−1.72, 2.34)

40
15
(25, 14 * 0)
(−1.56, 1.64)
(−1.87, 1.91)
(−1.29, 2.10)
(−1.49, 2.55)

40
15
(14 * 0, 25)
(−1.68, 1.56)
(−1.99, 1.83)
(−1.62, 1.80)
(−1.88, 2.19)

40
15
(10, 8, 7, 12 * 0)
(−1.68, 1.56)
(−1.99, 1.83)
(−1.61, 1.83)
(−1.88, 2.21)

40
20
(20, 19 * 0)
(−1.61, 1.66)
(−1.97, 1.92)
(−1.41, 1.98)
(−1.68, 2.37)

40
20
(19 * 0, 20)
(−1.69, 1.59)
(−2.05, 1.86)
(−1.63, 1.80)
(−1.95, 2.15)

40
20
(20 * 1)
(−1.64, 1.63)
(−2.02, 1.89)
(−1.51, 1.90)
(−1.80, 2.27)

50
20
(30, 19 * 0)
(−1.60, 1.67)
(−1.95, 1.92)
(−1.35, 2.04)
(−1.60, 2.44)

50
20
(19 * 0, 30)
(−1.69, 1.59)
(−2.05, 1.86)
(−1.63, 1.80)
(−1.95, 2.16)

50
20
(3 * 10, 17 * 0)
(−1.70, 1.59)
(−2.05, 1.85)
(−1.62, 1.82)
(−1.95, 2.17)

50
25
(25, 24 * 0)
(−1.61, 1.63)
(−1.94, 1.92)
(−1.43, 1.90)
(−1.68, 2.30)

50
25
(24 * 0, 25)
(−1.68, 1.58)
(−2.04, 1.84)
(−1.63, 1.75)
(−1.93, 2.09)

50
25
(25 * 1)
(−1.64, 1.61)
(−1.97, 1.88)
(−1.52, 1.83)
(−1.79, 2.21)

7. Real Data Analysis

In this section, we consider a real dataset to illustrate the inferential procedure developed in this article. The data studied by various authors are the number of the million revolutions before failure of the 25 ball bearings provided by Lieblein and Zelen [21]. The data are one of the most frequently used datasets in literature although incorrectly as a complete dataset, however, the data are indeed censored (see [22]).

Table 6 shows the ball bearings data. The six data values, as marked with asterisks, were discarded from analysis (censored). In Table 7, we presented the censored data as well as the progressively Type-II censoring scheme.

Table 6: Ball bearings data.

17.88
28.92
33.00
41.52
42.12
45.60
48.48
51.84
51.96

54.12
55.56
67.80
67.80∗
67.80∗
68.64
68.64∗
68.68∗
84.12

93.12
98.64
105.12
105.84∗
127.92
128.04
173.40∗

Table 7: Ball bearings data and progressively Type II censoring scheme.

1
2
3
4
5
6
7
8
9
10

17.88
28.92
33.00
41.52
42.12
45.60
48.48
51.84
51.96
54.12

0
0
0
0
0
0
0
0
0
0

11
12
13
14
15
16
17
18
19

55.56
67.80
68.64
84.12
93.12
98.64
105.12
127.92
128.04

2
0
2
0
0
0
1
0
1

The sample C.V. of the data is found to be 0.4917 and Table 1 suggests the value of the moment estimator of the shape parameter to be .

We found the approximate estimators for the parameters to be and . These estimates were then used as a starting values to find the MLEs from the system of normal equations (2.7) and (2.8) numerically. We obtained and . Figure 1 depicts the histogram of the data and the fitted GE-exponential .

Figure 1: Histogram of the ball bearings data and the density function of the fitted GE-exponential model.

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