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

Authors: Reza Pakyari
Journal: ISRN Probability and Statistics
 (2012)

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## 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.


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## Body

## 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. Supposen units are placed on test. At the time of the first failure, R1 units are randomly removed from the n-1 surviving units. Next, at the time of the second failure, R2 units are randomly removed from the n-R1-2 surviving units, and so on. Finally, after the mth failure, all remaining Rm units are removed. Thus, we observe m complete failures and R1+R2+⋯+Rm items are progressively censored from the n units under test, and so n=m+(R1+R2+⋯+Rm). The vector R=(R1,…,Rm) is called the censoring scheme and is fixed prior to the study. If R=(0,…,0), no withdrawals are made which correspond to the complete sample and the ordinary order statistics will be obtained. If R=(0,…,0,n-m), we obtain the conventional Type-II right censoring. We will denote the progressive Type-II censored data by X1:m:n<X2:m:n<⋯<Xm:m:n. 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 Section2. 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 parameterγ and scale parameter λ and with probability density function:(2.1)f(x;λ,γ)=γλe-λx(1-γ-e-λx)2,(x>0),
where γ-=1-γ. The cumulative distribution function is give by(2.2)F(x;λ,γ)=1-e-λx1-γ-e-λx.The GE-exponential reduces to the exponential distribution if we takeγ=1. When 0<γ<1, the distribution reduces to exponential geometric distribution studied by Adamidis and Loukas [14]. The GE-exponential distribution has decreasing failure rate when 0<γ<1 and increasing failure rate when γ≥1. The density function is unimodal with modal value equal to -λγ{4γ-}-1 when γ>2. Hereafter, we assume γ≥1.LetX1:m:n,…,Xm:m:n 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])
(2.3)L(λ,γ)=C∏i=1m‍γλe-λxi(1-γ-e-λxi)2{1-1-e-λxi1-γ-e-λxi}Ri=C(γλ)m∏i=1m‍γRie-λ(Ri+1)xi(1-γ-e-λxi)Ri+2,
where
(2.4)C=n(n-1-R1)(n-2-R1-R2)⋯(n-m+1-R1-⋯-Rm-1).
The log-likelihood function may be written as
(2.5)L*(λ,γ)=K+mlog⁡γ+mlog⁡λ+log⁡γ∑i=1m‍Ri-λ∑i=1m‍(Ri+1)xi-∑i=1m‍(Ri+2)log⁡(1-γ-e-λxi),
where K is a constant. Let zi=λxi, now the log-likelihood function may be rewritten as
(2.6)L*(λ,γ)=K+mlog⁡γ+mlog⁡λ+log⁡γ∑i=1m‍Ri-∑i=1m‍(Ri+1)zi-∑i=1m‍(Ri+2)log⁡(1-γ-e-zi).
The likelihood equations are then given by
(2.7)∂∂λL*(λ,γ)=mλ-1λ∑i=1m‍(Ri+1)zi-1λ∑i=1m‍(Ri+2)(γ-zie-zi1-γ-e-zi)=0,(2.8)∂∂γL*(λ,γ)=mγ+∑i=1mRiγ-∑i=1m‍(Ri+2)(e-zi1-γ-e-zi)=0,
which may be simplified as
(2.9)m-∑i=1m‍(Ri+1)zi-∑i=1m‍(Ri+2)(γ-zie-zi1-γ-e-zi)=0,(2.10)m+∑i=1m‍Ri-γ∑i=1m‍(Ri+2)(e-zi1-γ-e-zi)=0.
The MLEs of the parameters λ and γ 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 λ and γ 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 γ and based on this estimator finding an approximate MLE of the scale parameter λ and using that to find an approximate MLE of γ 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 γ is used to find the approximate maximum likelihood estimator of the scale parameter λ. The approximate MLE of γ is then can be found by either the likelihood equations (2.9) or (2.10) and the value of the approximate MLE of λ.To find a moment estimator ofγ, we consider the coefficient of variation (C.V.) which is given by
(3.1)C.V.=-γ-{2Φ(γ-,2,1)-(γln⁡2γ)/γ-2}1/2γln⁡γ,
where Φ(a,b,c) is the Lerch’s transcendent (see, for example, [18]).Note that C.V. is independent from the scale parameterλ and depends only on the shape parameter γ. Table 1 provides the value of the C.V. for different values of the shape parameter γ. This table may be used to find the moment estimator of the shape parameter γ 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 parameterγ for the GE-exponential distribution.
γ
C.V.
γ
C.V.
γ
C.V.
γ
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
0.408LetX1:m:n,…,Xm:m:n be a progressively Type-II censored sample from f(x;λ,γ~), where γ~ is the moment estimator of γ. The likelihood equation (2.9) may now be rewritten in terms of γ~ as
(3.2)m-∑i=1m‍(Ri+1)zi-∑i=1m‍(Ri+2)(γ~-zie-zi1-γ~-e-zi)=0,
where γ~-=1-γ~. The MLE of the scale parameter λ 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 h(z)=e-z/(1-γ~-e-z) in a Taylor series expansion around E(Zi:m:n)=νi:m:n. From Balakrishnan and Sandhu [19] we have
(3.3)F(Zi:m:n)=Ui:m:n,
where Ui:m:n is the ith order statistic of a progressively Type-II censored sample from the U(0,1) distribution.We then have(3.4)Zi:m:n=F-1(Ui:m:n)=log⁡(γ~-Ui:m:n-1Ui:m:n-1),
and hence
(3.5)νi:m:n=E(Zi:m:n)=log⁡(γ~-αi:m:n-1αi:m:n-1),
where αi:m:n=E(Ui:m:n) and
(3.6)αi:m:n=1-∏j=m-i+1m‍j+Rm-j+1+⋯+Rmj+1+Rm-j+1+⋯+Rm,i=1,…,m.
Expanding h(zi) around νi:m:n by Taylor series expansion and keeping the first two terms, we deduce
(3.7)h(zi)≈h(νi:m:n)+h′(zi)|z=νi:m:n(zi-νi:m:n)=αi+βizi,
where
(3.8)αi=e-νi:m:n(1-γ~-e-νi:m:n+νi:m:n)(1-γ~-e-νi:m:n)2,βi=-e-νi:m:n(1-γ~-e-νi:m:n)2.
Therefore, the likelihood equation (3.2) may be approximated by
(3.9)m-∑i=1m‍(Ri+1)zi-γ~-∑i=1m‍(Ri+2)zi(αi+βizi)=0.
Equation (3.9) may be rewritten as
(3.10){γ~-∑i=1m‍(Ri+2)βixi2}λ2+{γ~-∑i=1m‍(Ri+2)αixi+∑i=1m‍(Ri+1)xi}λ-m=0
or
(3.11)Aλ2+Bλ-m=0,
where
(3.12)A=γ~-∑i=1m‍(Ri+2)βixi2,B=γ~-∑i=1m‍(Ri+2)αixi+∑i=1m‍(Ri+1)xi.
Equation (3.11) is a quadratic equation in terms of λ with the following two solutions:(3.13)-B±B2+4Am2A.
However, one of the solutions is inadmissible since A>0. Hence, the approximate MLE of the scale parameter λ is given explicitly by
(3.14)λ˘=-B+B2+4Am2A.
The approximate MLE of γ, say γ˘ is then maybe found by plugging in the value of λ˘ in either (2.9) or (2.10) and solve the equation by a numerical method. Once the values of λ˘ and γ˘ 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 λ and γ.


## 4. Fisher Information

In this section, we derive the asymptotic variance-covariance matrix of the maximum likelihood estimators of the parametersλ and γ.The observed Fisher information may be obtained from the likelihood equations (2.7) and (2.8) as
(4.1)I=[I11I12I21I22],
where
(4.2)I11=-∂2∂λ2L*=mλ2-1λ2∑i=1m‍(Ri+2)γ-zi2e-zi(1-γ-e-zi)2,I12=I21=-∂2∂λ∂γL*=-1λ∑i=1m‍(Ri+2)zie-zi(1-γ-e-zi)2,I22=-∂2∂γ2L*=mγ2+∑‍i=1mRiγ2-∑i=1m‍(Ri+2)e-2zi(1-γ-e-zi)2.
The asymptotic variance-covariance matrix of the MLEs is then given by inverting the observed Fisher information matrix:
(4.3)V(λ^,γ^)=[I11I12I12I22]-1=[I11I12I12I22],
where
(4.4)I11=I22I11I22-I122,I12=-I12I11I22-I122,I22=I11I11I22-I122.
These quantities are estimated by evaluating them at MLEs
(4.5)I^11=I22I11I22-I122|λ=λ^,γ=γ^,I^12=-I12I11I22-I122|λ=λ^,γ=γ^,I^22=I11I11I22-I122|λ=λ^,γ=γ^
where λ^ and γ^ are the MLEs of λ and γ, respectively. We will use these asymptotic variances on Section 6 to derive asymptotic confidence intervals for the scale parameter λ and the shape parameter γ.


## 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 inR using the pseudorandom generator in that software package. We generated the progressively Type-II censored samples from the GE-exponential distribution with λ=1.0 and γ=3.0 using the algorithm presented by Balakrishnan and Sadhu [19]. The choice of γ=3.0 provides unimodality and increasing failure rate.Five values of the sample sizes, sayn=15,20,30,40,50, different choices of the effective sample size m, 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, (2*7,4*0) denotes the censoring scheme (7,7,0,0,0,0).We first calculated the moment estimator of the shape parameterγ using sample C.V. and Table 1. Then the approximate MLE of the scale parameter λ was calculated using (3.14). Approximate MLE of γ was then calculated by solving (2.8) numerically. The MLEs of λ and γ 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.Tables2 and 3 give the average of absolute bias, variance, and estimated variance calculated from observed Fisher information for MLEs λ^ and γ^, 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 λ^ and γ^. Also, as one would expect, both the bias and the variance decrease as the effective sample proportion that is, m/n 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 ofλ.
n
m
Scheme
AB(λ^)
var(λ^)
I
^
11
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
0.0309Table 3
Absolute bias, variance, and variance from observed Fisher information for the MLE ofγ.
n
m
Scheme
AB(γ^)
var(γ^)
I
^
22
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
0.0370The tables also show that for all sample sizesn and effective sample sizes m, the censoring scheme R=(n-m,0,…,0) tends to the smallest bias and variance, whilst the censoring scheme R=(0,0,…,n-m) 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 parametersλ and γ using the estimation for the observed Fisher information matrix. Note that the asymptotic distributions of the pivotal quantities
(6.1)P1=λ^-λI^11,P2=γ^-γI^22,
are standard normal distribution. The asymptotic 100(1-α)% confidence intervals for the parameters are then given by
(6.2)λ^±zα/2I^11,γ^±zα/2I^22,
where zα/2 is the upper α/2 critical value of the standard normal distribution.Table4 provides the coverage probabilities through 10000 Monte Carlo simulations for the parameters λ and γ. 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 parametersλ and γ.
n
m
Scheme
λ
γ
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
0.9448We also calculated the unconditional percentage points of these pivotal quantities and the results are given in Table5. The values of the percentage points associate with P1 and P2 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.
n
m
Scheme
P
1
P
2
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]).Table6 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.
i
1
2
3
4
5
6
7
8
9
10
x
i
:
m
:
n
17.88
28.92
33.00
41.52
42.12
45.60
48.48
51.84
51.96
54.12
R
i
0
0
0
0
0
0
0
0
0
0
i
11
12
13
14
15
16
17
18
19
x
i
:
m
:
n
55.56
67.80
68.64
84.12
93.12
98.64
105.12
127.92
128.04
R
i
2
0
2
0
0
0
1
0
1The sample C.V. of the data is found to be 0.4917 and Table1 suggests the value of the moment estimator of the shape parameter to be γ~=23.We found the approximate estimators for the parameters to beλ˘=0.0410 and γ˘=19.1628. 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 λ^=0.04 and γ^=17.67. Figure 1 depicts the histogram of the data and the fitted GE-exponential (0.04,17.67).Figure 1
Histogram of the ball bearings data and the density function of the fitted GE-exponential model.



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*Source: 292384-2012-09-11.xml*