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Estimation of the inverse Weibull parameters under adaptive type-II progressive hybrid censoring scheme
Accepted Manuscript Estimation of the inverse Weibull parameters under adaptive type-II progressive hybrid censoring scheme M. Nassar, O.E. Abo-Kasem PII: DOI: Reference:
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Received date: 12 August 2016 Revised date: 12 October 2016 Please cite this article as: M. Nassar, O.E. Abo-Kasem, Estimation of the inverse Weibull parameters under adaptive type-II progressive hybrid censoring scheme, Journal of Computational and Applied Mathematics (2016), http://dx.doi.org/10.1016/j.cam.2016.11.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Estimation of the Inverse Weibull Parameters Under Adaptive TypeII Progressive Hybrid Censoring Scheme M. Nassar and O.E. Abo-Kasem a
Department of Statistics, Faculty of Commerce, Zagazig University, Egypt.
ABSTRACT
This paper describes the frequentist and Bayesian estimation for the scale parameter and shape parameter of the inverse Weibull (IW) distribution based on adaptive type-II progressive hybrid censoring scheme (AT-II PHCS). We discuss the maximum likelihood estimators (MLEs) and the approximate MLEs, where the MLEs cannot be obtained in closed forms. The Bayes estimates for the IW parameters are obtained based on squared error (SE) loss function by using the approximation form of Lindley (1980). The optimal censoring scheme has been suggested using two different optimality criteria. A real life data set is used for illustration purpose. Finally, the different proposed estimators have been compared through an extensive simulation studies.
KEYWORDS: Inverse Weibull distribution; Adaptive type-II progressive hybrid censoring scheme;
Maximum
likelihood
estimation;
Bayesian
estimation;
Lindley
approximation.
1. Introduction If the random variable Y has a Weibull distribution, then the random variable
X Y
1
has an IW distribution with probability density function (pdf), cumulative
distribution function (cdf) and hazard rate function respectively given by
f (x ) x ( 1)e x , x 0, , 0,
(1)
and
F (x ) e x ,
(2)
and
h (x ) x ( 1) e x 1
1
1
.
The IW distribution is more appropriate model than the Weibull distribution because the Weibull distribution does not provide a satisfactory parametric fit if the data indicate a non-monotone and unimodal hazard rate functions. The hazard rate function of IW distribution can be decreasing or increasing depending on the value of the shape parameter. The IW distribution is useful to model several data such as the time to breakdown of an insulating fluid subjected to the action of a constant tension and degradation of mechanical components such as pistons and crankshafts of diesel engines. Extensive work has been done on the IW distribution, see for example, Keller and Kanath (1982), Erto and Rapone (1984), Calabria and Pulcini (1994), Maswadah (2003) and for more details about the generalizations of IW distribution see Oluyede and Yang (2014). In addition, many articles have considered IW distribution under different censoring schemes. Among others, Kundu and Howlader (2010), Musleh and Helu (2014), Sultan et al. (2014) and Xiuyun and Zaizai (2016).
Progressive hybrid censoring scheme in the context of life testing experiments was introduced by Kundu and Joarder. (2006). They considered a type-I progressive hybrid censoring scheme, in which n identical units are placed on test with predetermined progressive censoring scheme R1 , R 2 ,..., R m . At the time of the first failure x 1:m :n , R 1 units are randomly removed from the remaining n 1 surviving units. Similarly, at the time of the second failure x 2:m :n , R 2 units of the remaining n 2 R1 units are randomly removed and so on. The experiment is terminated at
random time T min x m :m :n , T where x m :m :n is the m -th failure and T (0, ) is a predetermined time. For more details see also Balakrishnan and Kundu (2013). The drawback of the type-I progressive hybrid censoring scheme is that the effective sample size is random and it can turn out to be a very small number, therefore, the statistical inference procedure may not be applicable or will have low efficiency. Ng et al. (2009) introduced an AT-II PHCS to increase the efficiency of statistical analysis and save the total test time and analyzed the data under the assumption of exponential distribution. In AT-II PHCS the effective number of failures m is fixed in advance and the experimental time is allowed to run over time T which is an ideal total test. In this case, the progressive censoring scheme R1 , R 2 ,..., R m is provided, but the values of some of the R i may be change accordingly during the experiment. If the 2
m-th progressively censored observed failures occurs before time T (i.e. X m :m :n T ), the experiment stops at this time X m :m :n , and we will have a usual type-II progressive censoring scheme with the pre-fixed progressive censoring scheme R1 , R 2 ,..., R m (Fig 1(a)). Otherwise, if X J :m :n T X J 1:m :n , where J 1 m and X J :m :n is the J-th failure time occur before time T, then we will not withdraw any items from the experiment by setting R J 1 , R J 2 ,..., R m 1 0 and at the time of the m-th failure all remaining surviving items are removed, i.e., R m n m i 1 R i (Fig 1(b)). The main J
advantage of this scheme is to speed up the experiment when the experiment duration exceed the predetermined time T and assures us to get the effective number of failures m. withdrawn
withdrawn
R1
X 1:m :n
withdrawn
R m 1
R2
X 2:m :n
withdrawn
. . . X m 1:m :n
Start
Rm
T
X m :m :n
End (a) Experiment terminates before time T.
withdrawn withdrawn
R1
X 1:m :n
R2
X 2:m :n
withdrawn
withdrawn n m i 1 R i J
RJ
. . . X J :m :n T
X J 1:m :n . . . X m 1:m :n X m :m :n
Start
End (b) Experiment terminates after time T.
Figure 1. Schematic representation of AT-II PHCS.
Many authors have considered AT-II PHCS. Lin et al. (2009), discussed the maximum likelihood and approximate maximum likelihood estimators for the Weibull distribution. Hemmati et al. (2013), studied the maximum likelihood and approximate maximum likelihood estimators for the log-normal distribution. Mahmoud et al. (2013), investigated the maximum likelihood and Bayes estimates of the unknown parameters of Pareto distribution. Ashour and Nassar (2014) showed the 3
MLE and asymptotic confidence intervals in the presence of competing risks. Ismail (2014), considered estimation problems of Weibull distribution under step-stress partially accelerated life test model. AL Sobhi and Soliman (2015), discussed the problem of estimating parameters of the exponentiated Weibull distribution. Hemmati and Khorram (2015), discussed the AT-II PHCS in the presence of competing risks with unknown causes of failures. This paper aims to investigate the classical and Bayesian estimation for the unknown parameters of the IW distribution based on AT-II PHCS. It is observed that the MLEs cannot be obtained in explicit forms and we suggest to use the approximate MLEs by expanding the normal equations using Taylor expansion. The Bayes estimates are obtained based on SE loss function under the assumption of independent gamma priors. The rest of this paper is organized as follows: In section 2, we discuss the maximum likelihood estimation. In section 3, we derive the approximate MLEs. Bayesian estimates under SE loss functions using Lindley approximation (1980) is provided in section 3. In section 4, Monte Carlo simulation results and the analysis of real data set are presented. We provide the optimal censoring scheme in Section 6. The paper is concluded in section 7.
2. Maximum Likelihood Estimation Let
x 1:m :n ... x J :m :n T x J 1:m :n ... x m :m :n
be an adaptive type- II
progressive censored sample of size m from IW distribution with pdf (1) and cdf (2) with censoring scheme R1 ,..., R J ,0,...,0, n m i 1 R i , then, the likelihood function J
ignoring the additive constant can be written as
L ( , ) e m
m
m
x i i 1
m
J
i 1
i 1
(1 e x m ) R m x i 1 (1 e x i ) R i
(3)
where x i x i:1m :n for simplicity of notation. The logarithm of the likelihood function
( ) is m
m
J
i 1
i 1
i 1
m ln( ) x i ( 1) ln(x i ) R i ln(1 e x i ) R m ln(1 e x m )
4
(4)
The MLEs of and can be obtained by deriving (4) with respect to and and equating the results to zero as follows J m m x i R i x i u i R m x m u m i 1 i 1
(5)
m m J m x i ln(x i ) ln(x i ) R i x i u i ln(x i ) R m x m u m ln(x m ) i 1 i 1 i 1
(6)
where u i e
x i
(1 e x i )1 (e x i 1)1 . It can be seen that equations (5) and (6)
cannot be solved explicitly, so, numerical methods are used to solve (5) and (6).
3. Approximate Maximum Likelihood Estimation (AMLE) It is noted that from (5) and (6) the MLE of and cannot be obtained in explicit form, we derive the AMLEs which have explicit forms. Let Z ln X , then Z
extreme value distribution with pdf and cdf given respectively by
g (x )
z z exp exp , z , 1
(7)
and
z G (x ) 1 exp exp ,
(8)
where ln / and 1/ are the location and scale parameters respectively. From (7) and (8) and under AT-II PHCS the log likelihood function can be written as follows
where t i
zi
m
m
J
i 1
i 1
i 1
m ln( ) t i e t i R i e t i R m e t m
(9)
, i 1,..., m . To obtain the AMLEs we expand the function e t i in
Taylor series around the points i F 1 (1 q i ) ln( ln(1 q i )), where
qi 1
m
j k m j 1 R k
j m i 1
j 1 k m j 1 R k
m
m
Keeping only the first order derivatives, we have e ti
i i t i ,
where 5
, i 1,..., m .
i e (1 i ) and i e 0. i
i
Using these linear approximations, we can obtain the approximate log likelihood equations as
m J 1 m ( t ) R i ( i i t i ) R m ( m m t m ) 0 , i i i i 1 i 1
and
m J 1 m t (1 ( t )) R i t i ( i i t i ) R t m ( m m t m ) 0 (10) i i i i i 1 i 1
The solutions of equations (10) are
ˆˆ A ˆˆ B and
D D 2 4mE ˆ ˆ 2m where m
A
J
i x i R i i x i R m m x m i 1
i 1 J
m
R i
i 1
m
B
i
i 1
J
R i
i 1
i
i 1
m
i
i 1
i
R m m
,
J
i
i
R mm m
R i 1
,
i
R m m
m
J
i 1
i 1
D ( i 1)(x i A ) R i i (x i A ) R m m (x m A ), and m
J
i 1
i 1
E i (x i A ) 2 R i i (x i A ) 2 R m m (x m A ) 2 . It is noted that the AMLEs have explicit solutions and they can be used as initial values to obtain the MLEs in (5) and (6).
6
4. Bayesian Estimation In this section, the SE loss function is used to obtain the Bayes estimates of the unknown parameters and . The Bayes estimates is considered under the assumption of independent gamma priors of and with the following joint density g , a 1 c 1e [b d ] ,
a, b , c , d 0
(11)
Based on the likelihood function (3) and the joint prior density (11), the joint posterior distribution of , and the data is
g , y m a 1
m x i b d m c 1 i 1
e
m
J
i 1
i 1
(1 e x m )R m x i 1 (1 e x i ) Ri
(12)
under SE loss function, the Bayes estimate of the function of the parameters and , sayU , is
0
0
U ( , )
U ( , ) L ( , ) g ( , ) d d
0
0
L ( , ) g ( , ) d d
0
0
U ( , ) e Q ( , ) .d .d
0
0
e Q ( , )d .d
(13)
where Q ( , ) ln L ( , ) ln g ( , ) (a,b ) . It is not possible to compute the ratio of the two integrals given by (13) in a closed form. Therefore, in such situation, we suggest using Lindley's approximation to obtain the Bayes estimates of the unknown parameters. For the two parameter case, ( , ) (1 , 2 ) , Lindley's approximation of (13) take the form 1 1 1 u ij ij U j j L30 11U 1 L 21 2 12U 1 11U 2 2 2 2 1 1 L12 22U 1 2 12U 2 L03 22U 2 , 2 2
U ( ) U ( )
(14)
where L
2U ( ) L ( ) U ( ) 3 and , , for i , j 1,2, u , , 0,1, 2,3 u ij i i j i 1 2
U k u i ki , j i
( ) , j 1,2 and j
ij is the (i,j)th element of the inverse of the
Fisher information matrix. Evaluation all functions in equation (14) at the MLE of
7
(1 , 2 ) , produces the approximation U ( ) to (13). Now, to apply Lindley's
approximation (14), we obtain the quantities
2 m J ˆ ˆ L20 2 2 R i x i2 i u i2 R m x m2 mu m2 , ˆ i 1 L02
J 2 m ˆ m ˆ 2 ˆ R x ˆ ln(x )2u 1 ˆx ˆ u x ln( x ) i i i i i i i i i 2 ˆ 2 i 1 i 1
ˆ ˆ ˆR m x m ln(x m )2u m 1 ˆx m mu m ,
L11
m J 2 ˆ ˆ ˆ x i ln(x i ) R i ˆx i2 ln(x i )i u i x iu i ln(x i ) i 1 i 1 ˆ ˆ R m ˆx m2 ln(x m )mu m x m u m ln(x m ) ,
L30
3 2m J ˆ ˆ R i x i3 i u i2 1 2i u i R m x m3 mu m2 1 2mu m , 3 3 ˆ i 1
L 03
J 3 2m ˆ m ˆ 3 ˆ R ln(x ) 2 x ˆu ln(x ) 1 ˆx ˆ u x ln( x ) i i i i i i i i i i 3 ˆ 3 i 1 i 1
ˆ ˆ ˆ ˆx i2 i u i2 ln(x i ) 2 ˆx i 2ˆx i i u i ˆR m ln(x m ) 2 ˆ ˆ ˆ x m u m ln(x m ) 1 ˆx m m u m ˆx m2 m u m2 ln(x m ) ˆ ˆ 2 ˆx m 2ˆx m m u m ,
L 21
2ˆx u 2ˆx u
J 3 ˆ R i x i2 ln(x i )i u i2 2 i 1
ˆ R m x m2 ln(x m )m u m2
ˆ
i
i
i
ˆ m
m
m
1 x
ˆ ˆ ˆx i 1 x i2 ln(x i )i u i2 ˆ ˆx m
2 ˆ m
ln(x m )m u m2 ,
and
m J 3 ˆ ˆ ˆ ˆ L12 2 x i ln(x i ) 2 R i ln(x i ) 2 x i ln(x i )u i 1 ˆx i i u i ˆx i2 i u i2 i 1 i 1
ˆ ˆ ˆ ˆ ˆ 2 ˆx i 2ˆx i i u i R m ln(x m ) 2 x m ln(x m )u m 1 ˆx m mu m ˆx m2 i u m2 ˆ ˆ 2 ˆx m 2ˆx m m u m ,
where i e as
x i
11
. The elements of the variance covariance matrix ij , can be obtained
L02 L20 L02 L112
, 22
L 20 L 20 L02 L112
Based on the joint prior function (11), we obtain
8
and 12 21
L11 L 20 L02 L112
.
( , ) ln g ( , ) (a 1)ln (c 1)ln b d hence 1
( , ) a 1 ( , ) c 1 b and 2 d .
If U ( , ) , u 1 1, u 2 0 and u11 u12 u 21 u 22 0 , then ˆ 1 11 2 12
L30 112 3L 21 11 12 L12 ( 11 22 2 122 ) L03 12 22 2
Similarly, when U ( , ) , hence, u 2 1, u 1 0 and u11 u12 u 21 u 22 0 . Thus ˆ 1 12 2 22
2 L30 11 12 L 21 2 122 11 22 3L12 22 12 L03 22
2
5. Data Analysis and Simulation Results The purpose of this section is to compare the performance of the different methods of estimation discussed in the previous sections. We analyze a real data set for illustrative purpose; also, a simulation study is carried out to check the behavior of the proposed methods as well as to evaluate the statistical performances of the estimators under different sampling schemes. 5.1. Real Data Example In this subsection, we reanalyze a real data set given by Blischke and Murthy (2000). The data set consists of 87 observations and it represents the failure times of aircraft windshields. Blischke and Murthy (2000) proved that the IW distribution gives a good fit for it. From the original data we generate three adaptive progressively hybrid censored samples with the following schemes Scheme
m
1
40
T 2.5
2
50
3
3
70
3.5
Censoring scheme
0*20,1*19, 28 1*25,0*24,12 17,0*69
the generated adaptive progressively hybrid censored samples given in the following table
9
Scheme 1
2
3
0.301
0.309
0.557
0.943
Censored data 1.070 1.124 1.248
1.281
1.281
1.303
1.480
1.505
1.506
1.568
1.615
1.619
1.652
1.652
1.757
1.795
1.866
1.899
1.911
1.981
2.010
2.085
2.089
2.097
2.154
2.194
2.223
2.300
2.349
2.481
2.625
2.632
2.661
2.823
2.902
2.934
0.301
0.309
0.557
0.943
1.124
1.248
1.281
1.281
1.432
1.480
1.505
1.568
1.615
1.652
1.652
1.795
1.866
1.876
1.899
1.911
1.912
1.981
2.038
2.085
2.135
2.154
2.194
2.224
2.229
2.324
2.385
2.610
2.632
2.646
2.688
2.890
2.934
2.962
3.000
3.114
3.117
3.344
3.376
3.385
3.467
3.478
3.578
3.595
3.699
3.779
0.301
0.309
0.557
0.943
1.070
1.248
1.281
1.281
1.432
1.480
1.505
1.506
1.615
1.619
1.652
1.652
1.757
1.795
1.866
1.876
1.899
1.911
1.912
1.914
1.981
2.038
2.085
2.089
2.097
2.135
2.154
2.190
2.194
2.223
2.229
2.324
2.349
2.481
2.610
2.625
2.632
2.646
2.661
2.688
2.823
2.890
2.934
2.962
3.000
3.103
3.117
3.344
3.376
3.385
3.467
3.478
3.578
3.595
3.779
4.035
4.121
4.240
4.255
4.278
4.305
4.449
4.485
4.570
4.602
4.694
The MLEs, AMLEs and Bayes estimates using Lindley's approximation are reported in table 1. The Bayes estimates of and are computed based on a non-informative prior, i.e., a b c d 0 . From table 1, it is noted that the MLEs and Bayes estimates of the unknown parameters are quite close to each other. Also, the estimates based on adaptive progressively hybrid type-II samples are very close to those of complete data and this demonstrate the importance and usefulness of AT-II PHCS. Table 1: MLEs, AMLEs and Bayes estimates of and . Scheme Complete 1 2 3
2.157 2.335 2.288 1.984
MLEs
AMLEs
1.392 0.965 1.128 1.353
282.762 68.277 33.14 102.357
4.277 4.086 2.799 3.550
Bayes estimates
2.149 2.325 2.279 1.975
1.387 0.957 1.121 1.348
5.2. Simulation Study This subsection aims to conduct a Mont Carlo simulation study to compare the performance of the different methods of estimation proposed in the previous sections. The performance of all the estimates is compared in terms of their mean square errors (MSEs). The average values and MSEs of MLEs, AMLEs and Bayes estimates of and are evaluated for , 0.5,1 and , 0.75, 2 and by considering different values n , m , T and three censoring schemes as follows 10
Scheme 1: R1 · · · R m 1 0 and R m n m , Scheme 2: R1 · · · R m 1 1 and R m n 2m 1 , and Scheme 3: R1 · · · R m 1 = R m (n m ) / m . We replicate the process 1000 times. The average values and MSEs are tabulated in tables 3-6 . The Bayes estimates are all computed under non-informative and informative priors. The priors for the two sets of parameters are given in table 2. We use prior 0 for the two sets of parameter values, Prior 1 for , 0.5,1 and prior 2 for , 0.75, 2 . From table 2, it may be noted that prior 1 describes the case of non-informative prior and prior 1 and prior 2 are chosen in such way that prior means equal to the original means. Table 2: Different priors for the two sets of parameters of and . Parameters Prior ( 0.5, 1) and ( 0.75, 2) ( 0.5, 1) ( 0.75, 2)
Prior 0: a b c d 0. Prior 1: a 1, b , 2, c 1, d 1. Prior 2: a 3, b , 4, c 4, d 2.
From tables 3-6, it can be seen that the Bayes estimates using prior 1 perform better than those by using AMLEs, MLEs and Bayes estimates using prior 0 in terms of minimum MSEs for , 0.5,1 , while the AMLEs has the minimum MSEs among all other methods for , 0.75, 2 . It is observed that the Bayes estimates with respect to the non-informative prior (prior 0) are quite close to the MLEs. For fixed n and T , when m increases the MSEs of MLEs, AMLEs and Bayes estimates decrease except come cases. For fixed n and m , when T increases we do not observe any specific trend in the MSEs. In most cases, for the parameter , the MSEs of scheme 3 are always smaller than other schemes, while, for the parameter the scheme 1 has the minimum MSEs, except some cases.
11
Table 3: The average values and MSEs of the parameters , 0.5,1 under different censoring schemes and for T 0.5 . (n , m )
Scheme
Parameter
AML Estimates
ML Estimates
(30,10)
1
0.84120 0.12074 1.38931 0.16330
0.62652 0.08001 1.01470 0.10278
Prior 0 0.67608 0.09109 1.01639 0.09627
Prior 1 0.63785 0.06194 1.00692 0.06646
0.80237 0.09563 1.43781 32891.0
0.55475 0.08011 1.05662 0.11566
0.59505 0.08183 1.02897 0.10672
0.57089 0.05796 1.03481 0.07282
0.76593 0.07328 1.48584 0.24096
0.50452 0.07075 1.12891 0.15532
0.53115 0.06965 1.10470 0.14096
0.51538 0.04991 1.09623 0.09344
0.89938 0.16010 1.46719 0.22143
0.72228 0.12108 1.03026 0.08340
0.79207 0.15587 1.06119 0.08097
0.72570 0.09724 1.01944 0.05013
0.88584 0.14959 1.49804 0.25100
0.68010 0.09758 1.01656 0.08649
0.74499 0.12429 1.04696 0.08307
0.38939 0.07874 1.01312 0.05179
0.83352 0.11231 1.56630 0.32316
0.55118 0.06373 1.18009 0.18966
0.50708 0.06171 1.15314 0.17083
0.51742 0.04452 1.12570 0.09651
0.87597 0.14202 1.45344 0.20808
0.74573 0.10340 1.08966 0.05249
0.78807 0.12475 1.11037 0.05396
0.74330 0.09172 1.07632 0.04199
0.85861 0.12949 1.49069 0.24306
0.65411 0.06174 1.04743 0.04983
0.69224 0.07405 1.06676 0.05005
0.66041 0.05486 1.04078 0.03940
0.80155 0.09195 1.54896 0.30328
0.50084 0.04940 1.07197 0.09662
0.52877 0.04966 1.05670 0.09179
0.51678 0.03961 1.05788 0.06853
0.83662 0.11570 1.41143 0.17351
0.66112 0.05742 1.07117 0.04028
0.68658 0.06507 1.08558 0.04134
0.66317 0.05247 1.06426 0.03535
0.79318 0.08829 1.45456 0.20926
0.57892 0.05319 1.02938 0.05373
0.59918 0.05519 1.04213 0.05295
0.58438 0.04592 1.02974 0.04678
0.75834 0.06810 1.50145 0.25351
0.52652 0.04541 1.02493 0.05805
0.54495 0.04580 1.01455 0.05638
0.53430 0.03829 1.02196 0.04933
2
3
(50,10)
1
2
3
(60,15)
1
2
3
(60,20)
1
2
3
12
Bayes Estimates
Table 4: The average values and MSEs of the parameters , 0.5,1 under different censoring schemes and for T 1 . (n , m )
Scheme
Parameter
AML Estimates
ML Estimates
(30,10)
1
0.84926 0.12332 1.39117 0.15957
0.73272 0.11794 1.04086 0.08473
Prior 0 0.78493 0.14242 1.07073 0.08296
Prior 1 0.72916 0.09645 1.02981 0.05671
0.82409 0.10653 1.44180 0.20065
0.60422 0.05799 1.02692 0.08288
0.65013 0.06810 1.00187 0.07749
0.61638 0.04571 1.02223 0.05450
0.78353 0.08306 1.48678 0.24150
0.51580 0.06699 1.31944 0.32484
0.58024 0.06029 1.29266 0.29855
0.58026 0.04532 1.23237 0.16178
0.89989 0.16050 1.46482 0.21908
0.71648 0.11728 1.02301 0.07936
0.78650 0.15136 1.05425 0.07642
0.72134 0.09437 1.01192 0.04864
0.88556 0.14932 1.49425 0.24722
0.69129 0.10087 1.01951 0.08411
0.75653 0.12912 1.04985 0.08086
0.69904 0.08178 1.01299 0.04973
0.84124 0.11734 1.56069 0.31697
0.63707 0.05350 1.28768 0.22532
0.59261 0.04442 1.25882 0.20335
0.58789 0.03295 1.21168 0.11488
0.87473 0.14104 1.44579 0.20162
0.77594 0.12545 1.09747 0.05273
0.81770 0.14904 1.11672 0.05500
0.76998 0.11080 1.07996 0.04232
0.85548 0.12704 1.48228 0.23530
0.72880 0.09645 1.08661 0.05005
0.76672 0.11421 1.10513 0.05186
0.72682 0.08560 1.07331 0.04064
0.81379 0.09948 1.54321 0.29723
0.60824 0.04596 1.22298 0.15309
0.58022 0.04095 1.20485 0.14204
0.58037 0.03381 1.18671 0.10617
0.83568 0.11345 1.40612 0.16856
0.81355 0.13249 1.14773 0.04968
0.83929 0.14835 1.16119 0.05294
0.80397 0.12113 1.13289 0.04320
0.80755 0.09546 1.45627 0.21124
0.69775 0.06476 1.10287 0.03802
0.72027 0.07353 1.11557 0.03999
0.69518 0.05949 1.09319 0.03346
0.76361 0.07080 1.50307 0.25555
0.54139 0.03354 1.11271 0.08441
0.52242 0.03203 1.10111 0.08013
0.52791 0.02736 1.10137 0.06772
2
3
(50,10)
1
2
3
(60,15)
1
2
3
(60,20)
1
2
3
13
Bayes Estimates
Table 5: The average values and MSEs of the parameters , 0.75, 2 under different censoring schemes and for T 1 . (n , m )
Scheme
Parameter
AML Estimates
ML estimates
(30,10)
1
0.80937 0.00670 2.40940 0.27804
0.97243 0.12133 2.02838 0.32448
Prior 0 1.01723 0.13846 2.08998 0.31099
Prior 2 0.96578 0.07229 2.09384 0.09764
0.77352 0.00463 2.56589 0.39789
0.82462 0.06817 2.10313 0.37125
0.86329 0.07100 2.04529 0.34155
0.85689 0.03292 2.05653 0.09090
0.77801 0.00422 2.70167 0.54953
0.81926 0.09084 2.40655 0.92012
0.78693 0.08408 2.35887 0.85149
0.75430 0.03356 2.02725 0.19757
0.87252 0.01585 2.63341 0.43911
1.05224 0.18545 2.08213 0.34465
1.11549 0.22284 2.14326 0.33693
1.02722 0.10679 2.10348 0.09149
0.85626 0.01219 2.72912 0.56563
1.02066 0.16279 2.08233 0.32825
1.07942 0.19344 2.14169 0.32144
1.00374 0.09451 2.11278 0.09292
0.80559 0.00435 2.94120 0.91059
0.86882 0.08934 2.44654 0.79588
0.82355 0.07901 2.38578 0.72915
0.75794 0.03493 2.01157 0.78184
0.84135 0.00912 2.56649 0.35271
1.07890 0.17779 2.19988 0.20610
1.13491 0.20186 2.23831 0.21579
1.06367 0.12897 2.15770 0.11154
0.81974 0.00568 2.68623 0.49808
1.04386 0.13741 2.18700 0.19983
1.07635 0.15549 2.22380 0.20813
1.01778 0.09975 2.16041 0.11244
0.76626 0.00144 2.88341 0.79959
0.79554 0.06345 2.22991 0.39298
0.76933 0.06002 2.19805 0.36808
0.75069 0.03192 2.07146 0.12119
0.79880 0.00368 2.46771 0.25553
1.05816 0.13076 2.22537 0.15954
1.07993 0.14337 2.25353 0.16944
1.03358 0.10414 2.19499 0.11358
0.76764 0.00222 2.62354 0.41512
0.86429 0.04151 2.05494 0.13237
0.88375 0.04522 2.08244 0.13210
0.86827 0.03185 2.07294 0.08740
0.78881 0.00281 2.74665 0.57809
0.77950 0.03809 2.15901 0.21259
0.76286 0.03636 2.13672 0.20164
0.75235 0.02327 2.08142 0.10755
2
3
(50,10)
1
2
3
(60,15)
1
2
3
(60,20)
1
2
3
14
Bayes Estimates
Table 6: The average values and MSEs of the parameters , 0.75, 2 under different censoring schemes and for T 1.5 . (n , m )
Scheme
Parameter
AML Estimates
ML estimates
(30,10)
1
0.80063 0.00484 2.35464 0.24384
1.08155 0.19673 2.11599 0.35206
Prior 0 1.12358 0.22080 2.17462 0.34848
Prior 2 1.04211 0.12138 2.13630 0.11575
0.76925 0.00269 2.52477 0.36646
0.95569 0.11513 2.02937 0.32898
0.99240 0.12717 2.08430 0.31716
0.94513 0.06750 2.09950 0.10404
0.76182 0.00339 2.69557 0.53999
0.92067 0.10518 2.67367 1.29705
0.88585 0.09175 2.61847 1.19010
0.80097 0.03047 2.04240 0.18736
0.87179 0.01571 2.62097 0.42686
1.06107 0.19045 2.07749 0.34702
1.12404 0.22894 2.13861 0.33905
1.03381 0.11103 2.10085 0.09573
0.80555 0.01207 2.72037 0.55510
1.02722 0.16575 2.07678 0.33217
1.08566 0.19734 2.13629 0.32462
1.00775 0.09785 2.10681 0.11298
0.80381 0.00396 2.93084 0.89270
0.86408 0.06718 2.42988 0.65334
0.81913 0.05780 2.37646 0.59091
0.75530 0.01844 2.04042 0.06849
0.84097 0.00917 2.56394 0.35049
1.09847 0.18361 2.19213 0.20527
1.08445 0.20742 2.23079 0.21443
1.06455 0.13316 2.15411 0.11037
0.84087 0.00914 2.56455 0.35102
1.10032 0.18444 2.19536 0.20795
1.09626 0.20836 2.23396 0.21721
1.06585 0.13353 2.15630 0.11316
0.77329 0.00159 2.87405 0.78316
0.86897 0.05540 2.38978 0.45160
0.84140 0.04869 2.35430 0.41630
0.80016 0.02330 2.17473 0.12152
0.79119 0.00280 2.41656 0.21571
1.08908 0.18348 2.27255 0.18525
1.14962 0.19805 2.29980 0.19727
1.09439 0.14591 2.23131 0.13355
0.75871 0.00121 2.58340 0.37295
1.01324 0.10422 2.21404 0.16030
1.03098 0.11268 2.23927 0.16840
0.99195 0.08199 2.19336 0.11776
0.78150 0.00246 2.74098 0.57189
0.83505 0.04505 2.30239 0.34530
0.81766 0.04146 2.27811 0.32513
0.79505 0.02512 2.17909 0.15389
2
3
(50,10)
1
2
3
(60,15)
1
2
3
(60,20)
1
2
3
15
Bayes Estimates
6. Optimal Censoring Scheme In practice, it is very important to choose the optimum censoring scheme to obtain the highest amount of information about the unknown parameters. In the case of AT-II PHCS, for fixed n and m , the practitioner might be interested to choose the ideal time T and the progressive censoring scheme R1 ,..., R m , where n m i 1 R i m
and R J 1 , R J 2 ,..., R m 1 0 if X J :m :n T X J 1:m :n , which provides more information about the unknown parameters under consideration. Here, we investigate the optimal censoring schemes in terms of (a) Criteria 1: minimum trace of the variancecovariance matrix of the MLEs ( ) and (b) Criteria 2: minimizing the variance of the estimate of the p-th quantile given by
V (Qˆ p ) V 11 (ˆ) F( p1) V 22 (ˆ) 2F(11p )V 12 (ˆ) 2
where Qˆ p is the MLE of the p-th quintile and F( p1) is the inverse CDF of the IW distribution defined as 1 (p)
F
ln( p )
1/
Illustrative Example: Using the real data set of Blischke and Murthy (2000), We generate an adaptive progressively hybrid censored samples considering different values of m , T and censoring schemes. The comparison results of different optimal censoring schemes are presented in tables 7-10 by obtaining and the variance of 5-th quantiles V (Qˆ0.05 ) and 95-th quantiles V (Qˆ0.95 ) .
Table 7: Different censoring planes for m 30 and T 2 . Scheme
ˆ
ˆ
0*29,57
2.3080
0*20,1*9, 48 1*29, 28 2*10,1*10,0*9, 27 1*20, 2*9,19 57,0*29 0*10,1*10,0*9, 47
0.9320
V (Qˆ0.05 ) 0.6870
V (Qˆ0.95 ) 48.3870
0.0820
2.3170
0.9220
0.7404
52.4120
0.0821
2.4009
0.8718
1.1089
80.881
0.0894
2.3683
0.8963
0.8726
63.136
0.0896
2.3599
0.9122
0.8193
56.501
0.0868
1.3827
1.2417
0.2051
4.2986
0.0806
2.3654
0.9044
0.8865
61.6079
0.0851
Table 8: Different censoring planes for m 30 and T 2.5 . 16
Scheme
ˆ
ˆ
0*29,57
2.3080
0*20,1*9, 48 1*29, 28 2*10,1*10,0*9, 27 1*20, 2*9,19 57,0*29 0*10,1*10,0*9, 47
0.9320
V (Qˆ0.05 ) 0.6870
V (Qˆ0.95 ) 48.3870
0.0820
2.3890
0.9020
0.9680
64.0130
0.0870
2.4390
0.8490
1.3820
101.8880
0.0920
2.5140
0.8120
1.8590
148.9720
0.1000
2.4920
0.8150
1.8820
147.4520
0.0960
1.9755
1.3453
0.2105
5.3845
0.1522
2.3450
0.9140
0.8060
56.4180
0.0840
V (Qˆ0.95 ) 15.1367
0.0739
Table 9: Different censoring planes for m 40 and T 2 . Scheme
ˆ
ˆ
0*39, 47
2.2342
1.0629
V (Qˆ0.05 ) 0.3977
0*20,1*19, 28 1*39,8
2.1421
1.1307
0.2224
9.8835
0.0714
2.3354
1.018
0.5452
20.4027
0.0821
2*15,1*10,0*14,7 1*9, 2*19,0*12 47,0*39 0*15,1*15,0*9,32
2.5324
0.9137
1.1716
46.9090
0.0958
2.4130
1.0077
0.6174
22.4141
0.0873
1.5517
1.2985
0.0443
3.3231
0.0759
2.2898
1.0057
0.5575
21.9797
0.0770
Table 10: Different censoring planes for m 40 and T 2.5 . Scheme
ˆ
ˆ
0*39, 47
2.2342
0*20,1*19, 28 1*39,8 2*15,1*10,0*14,7 1*9, 2*19,0*12 47,0*39 0*15,1*15,0*9,32
1.0629
V (Qˆ0.05 ) 0.3977
V (Qˆ0.95 ) 15.1367
0.0739
2.2612
1.0248
0.4949
19.2707
0.0753
2.2750
1.0316
0.4437
18.1368
0.0785
2.3690
1.0263
0.4804
19.2229
0.0870
2.3535
1.0211
0.4702
19.9133
0.0836
1.6393
1.3048
0.0068
3.4906
0.0827
2.2893
0.9886
0.6055
24.6367
0.0769
From tables 7-10, it is noted that the censoring scheme number six has the minimum variance of the estimate of the p-th quantile, i.e. V (Qˆ0.05 ) and V (Qˆ0.95 ) .
17
7. Conclusion In this paper, we discussed the estimation problem of the unknown parameters of the IW distribution based on adaptive type-II progressively hybrid censored data. We used classical and Bayesian estimation methods to estimate the unknown parameters. It is observe that the MLEs cannot be obtained in explicit forms, therefore, the AMLEs have been suggested. We obtained the Bayes estimates based on SE loss function under the assumption of independent gamma priors using Lindley's approximation. A real data set is used to show how the scheme works in practice. The performance of the different estimators is compared based on simulation study in terms of their MSEs. It is observed that the Bayes estimates with non-informative priors work very well in terms of MSEs, also, the Bayes estimates using the informative prior have the smallest MSEs among all other estimates proposed. Finally, we have proposed the optimal censoring scheme based on two different criteria and an illustrative example is provided. As a future work, the inferential results of some life time models under adaptive type-II progressively hybrid censored data with random removals have not been developed yet. Also, one sample prediction based on AT-II PHCS remains open.
Acknowledgments The authors thank the anonymous referee for a careful reading of the article.
References AL Sobhi, M. M. and Soliman, A.A. (2015). Estimation for the exponentiated Weibull model with adaptive Type-II progressive censored schemes. Applied Mathematical Modelling. 40 (2), 1180-192. Ashour, S.K. and Nassar, M. M. A. (2014). Analysis of generalized exponential distribution under adaptive Type-II progressive hybrid censored competing risks data. International Journal of Advanced Statistics and Probability.2, 108-113. Balakrishnan, N. and Kundu, D. (2013). Hybrid censoring: models, inferential results and applications. Computational Statistics and Data Analysis. 57 (1), 166 - 209. Blischke, W.R. and Murthy, D.N.P. (2000). Reliability: Modeling, Prediction, and Optimization. New York: Wiley.
18
Calabria, R., Pulcini, G. (1994). Bayesian 2-sample prediction for the inverse Weibull distribution. Communications in Statistics-Theory and Methods. 23, 1811-1824. Erto, P., Rapone, M. (1984). Non-informative and practical Bayesian confidence bounds for reliable life in the Weibull model. Reliability Engineering. 7, 181-191. Hemmati, F. and Khorram, E. (2013). Statistical analysis of the log-normal distribution under type- II progressive hybrid censoring schemes. Communications in Statistics-Simulation and Computation. 42, 52-75. Hemmati, F. and Khorram, E. (2015) .On adaptive progressively type-II censored competing risks data. Communications in Statistics-Simulation and Computation. (to appear). Ismail, A.A. (2014), Inference for a step-stress partially accelerated life test model with an adaptive Type-II progressively hybrid censored data from Weibull distribution. Journal of Computational and Applied Mathematics. 260, 553-542. Keller, A. Z, Kamath,A. R. R. (1982). Alternative reliability models for mechanical systems. In: Proceeding of the third international conference on reliability and maintainability. 411-415. Kundu, D. and Howlader, H. (2010). Bayesian inference and prediction of the inverse Weibull distribution for type-II censored data. Computational Statistics and Data Analysis. 54, 1547-1558. Kundu, D. and Joarder, A. (2006). Analysis of type-II progressively hybrid censored data. Computational Statistics and Data Analysis. 50, 2509-2528. Lin, C. T., Ng, H. K. T and Chan, P. S. (2009). Statistical inference of type-II progressively hybrid censored data with Weibull lifetimes. Communications in Statistics-Theory and Methods. 38, 1710- 1729. Lindley, D.V. (1980). Approximate Bayesian methods . Trabajos de Estadistica. 31, 223-237. Mahmoud, M. A. W., Soliman, A. A., Abd Ellah, A. H. and El-Sagheer, R .M. (2013). Estimation of generalized Pareto under an adaptive type-II progressive censoring. Intelligent Information Management. 5, 73-83. Maswadah, M. (2003). Conditional confidence interval estimation for the inverse Weibull distribution based on censored generalized order statistics. Journal of Statistical Computation and Simulation. 73(12), 887-898. Musleh, R. and Helu, A. (2014).Estimation of the inverse Weibull distribution based on progressively censored data: Comparative study. Reliability Engineering and System Safety. 131, 216-227.
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Ng, H.K.T., Kundu, D. and Chan, P.S. (2009). Statistical analysis of exponential lifetimes under an adaptive Type-II progressively censoring scheme. Naval Research Logistics, 56, 687-698. Oluyede, B.O. and Yang, T. (2014). Generalizations of the inverse Weibull and related distributions with applications. Electronic Journal of Applied Statistical Analysis . 7, 94-116. Sultan, K. S., Alsadat, N. H. and Kundu, D. (2014).Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-II censoring. Journal of Statistical Computation and Simulation. 84 (10), 2248-2265. Xiuyun, P. and Zaizai, Y. (2016). Bayesian estimation and prediction for the inverse weibull distribution under general progressive censoring. Communications in Statistics-Theory and Methods. 45, 624-635.
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S0377-0427(16)30545-3 http://dx.doi.org/10.1016/j.cam.2016.11.012 CAM 10885
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Journal of Computational and Applied Mathematics
Received date: 12 August 2016 Revised date: 12 October 2016 Please cite this article as: M. Nassar, O.E. Abo-Kasem, Estimation of the inverse Weibull parameters under adaptive type-II progressive hybrid censoring scheme, Journal of Computational and Applied Mathematics (2016), http://dx.doi.org/10.1016/j.cam.2016.11.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Estimation of the Inverse Weibull Parameters Under Adaptive TypeII Progressive Hybrid Censoring Scheme M. Nassar and O.E. Abo-Kasem a
Department of Statistics, Faculty of Commerce, Zagazig University, Egypt.
ABSTRACT
This paper describes the frequentist and Bayesian estimation for the scale parameter and shape parameter of the inverse Weibull (IW) distribution based on adaptive type-II progressive hybrid censoring scheme (AT-II PHCS). We discuss the maximum likelihood estimators (MLEs) and the approximate MLEs, where the MLEs cannot be obtained in closed forms. The Bayes estimates for the IW parameters are obtained based on squared error (SE) loss function by using the approximation form of Lindley (1980). The optimal censoring scheme has been suggested using two different optimality criteria. A real life data set is used for illustration purpose. Finally, the different proposed estimators have been compared through an extensive simulation studies.
KEYWORDS: Inverse Weibull distribution; Adaptive type-II progressive hybrid censoring scheme;
Maximum
likelihood
estimation;
Bayesian
estimation;
Lindley
approximation.
1. Introduction If the random variable Y has a Weibull distribution, then the random variable
X Y
1
has an IW distribution with probability density function (pdf), cumulative
distribution function (cdf) and hazard rate function respectively given by
f (x ) x ( 1)e x , x 0, , 0,
(1)
and
F (x ) e x ,
(2)
and
h (x ) x ( 1) e x 1
1
1
.
The IW distribution is more appropriate model than the Weibull distribution because the Weibull distribution does not provide a satisfactory parametric fit if the data indicate a non-monotone and unimodal hazard rate functions. The hazard rate function of IW distribution can be decreasing or increasing depending on the value of the shape parameter. The IW distribution is useful to model several data such as the time to breakdown of an insulating fluid subjected to the action of a constant tension and degradation of mechanical components such as pistons and crankshafts of diesel engines. Extensive work has been done on the IW distribution, see for example, Keller and Kanath (1982), Erto and Rapone (1984), Calabria and Pulcini (1994), Maswadah (2003) and for more details about the generalizations of IW distribution see Oluyede and Yang (2014). In addition, many articles have considered IW distribution under different censoring schemes. Among others, Kundu and Howlader (2010), Musleh and Helu (2014), Sultan et al. (2014) and Xiuyun and Zaizai (2016).
Progressive hybrid censoring scheme in the context of life testing experiments was introduced by Kundu and Joarder. (2006). They considered a type-I progressive hybrid censoring scheme, in which n identical units are placed on test with predetermined progressive censoring scheme R1 , R 2 ,..., R m . At the time of the first failure x 1:m :n , R 1 units are randomly removed from the remaining n 1 surviving units. Similarly, at the time of the second failure x 2:m :n , R 2 units of the remaining n 2 R1 units are randomly removed and so on. The experiment is terminated at
random time T min x m :m :n , T where x m :m :n is the m -th failure and T (0, ) is a predetermined time. For more details see also Balakrishnan and Kundu (2013). The drawback of the type-I progressive hybrid censoring scheme is that the effective sample size is random and it can turn out to be a very small number, therefore, the statistical inference procedure may not be applicable or will have low efficiency. Ng et al. (2009) introduced an AT-II PHCS to increase the efficiency of statistical analysis and save the total test time and analyzed the data under the assumption of exponential distribution. In AT-II PHCS the effective number of failures m is fixed in advance and the experimental time is allowed to run over time T which is an ideal total test. In this case, the progressive censoring scheme R1 , R 2 ,..., R m is provided, but the values of some of the R i may be change accordingly during the experiment. If the 2
m-th progressively censored observed failures occurs before time T (i.e. X m :m :n T ), the experiment stops at this time X m :m :n , and we will have a usual type-II progressive censoring scheme with the pre-fixed progressive censoring scheme R1 , R 2 ,..., R m (Fig 1(a)). Otherwise, if X J :m :n T X J 1:m :n , where J 1 m and X J :m :n is the J-th failure time occur before time T, then we will not withdraw any items from the experiment by setting R J 1 , R J 2 ,..., R m 1 0 and at the time of the m-th failure all remaining surviving items are removed, i.e., R m n m i 1 R i (Fig 1(b)). The main J
advantage of this scheme is to speed up the experiment when the experiment duration exceed the predetermined time T and assures us to get the effective number of failures m. withdrawn
withdrawn
R1
X 1:m :n
withdrawn
R m 1
R2
X 2:m :n
withdrawn
. . . X m 1:m :n
Start
Rm
T
X m :m :n
End (a) Experiment terminates before time T.
withdrawn withdrawn
R1
X 1:m :n
R2
X 2:m :n
withdrawn
withdrawn n m i 1 R i J
RJ
. . . X J :m :n T
X J 1:m :n . . . X m 1:m :n X m :m :n
Start
End (b) Experiment terminates after time T.
Figure 1. Schematic representation of AT-II PHCS.
Many authors have considered AT-II PHCS. Lin et al. (2009), discussed the maximum likelihood and approximate maximum likelihood estimators for the Weibull distribution. Hemmati et al. (2013), studied the maximum likelihood and approximate maximum likelihood estimators for the log-normal distribution. Mahmoud et al. (2013), investigated the maximum likelihood and Bayes estimates of the unknown parameters of Pareto distribution. Ashour and Nassar (2014) showed the 3
MLE and asymptotic confidence intervals in the presence of competing risks. Ismail (2014), considered estimation problems of Weibull distribution under step-stress partially accelerated life test model. AL Sobhi and Soliman (2015), discussed the problem of estimating parameters of the exponentiated Weibull distribution. Hemmati and Khorram (2015), discussed the AT-II PHCS in the presence of competing risks with unknown causes of failures. This paper aims to investigate the classical and Bayesian estimation for the unknown parameters of the IW distribution based on AT-II PHCS. It is observed that the MLEs cannot be obtained in explicit forms and we suggest to use the approximate MLEs by expanding the normal equations using Taylor expansion. The Bayes estimates are obtained based on SE loss function under the assumption of independent gamma priors. The rest of this paper is organized as follows: In section 2, we discuss the maximum likelihood estimation. In section 3, we derive the approximate MLEs. Bayesian estimates under SE loss functions using Lindley approximation (1980) is provided in section 3. In section 4, Monte Carlo simulation results and the analysis of real data set are presented. We provide the optimal censoring scheme in Section 6. The paper is concluded in section 7.
2. Maximum Likelihood Estimation Let
x 1:m :n ... x J :m :n T x J 1:m :n ... x m :m :n
be an adaptive type- II
progressive censored sample of size m from IW distribution with pdf (1) and cdf (2) with censoring scheme R1 ,..., R J ,0,...,0, n m i 1 R i , then, the likelihood function J
ignoring the additive constant can be written as
L ( , ) e m
m
m
x i i 1
m
J
i 1
i 1
(1 e x m ) R m x i 1 (1 e x i ) R i
(3)
where x i x i:1m :n for simplicity of notation. The logarithm of the likelihood function
( ) is m
m
J
i 1
i 1
i 1
m ln( ) x i ( 1) ln(x i ) R i ln(1 e x i ) R m ln(1 e x m )
4
(4)
The MLEs of and can be obtained by deriving (4) with respect to and and equating the results to zero as follows J m m x i R i x i u i R m x m u m i 1 i 1
(5)
m m J m x i ln(x i ) ln(x i ) R i x i u i ln(x i ) R m x m u m ln(x m ) i 1 i 1 i 1
(6)
where u i e
x i
(1 e x i )1 (e x i 1)1 . It can be seen that equations (5) and (6)
cannot be solved explicitly, so, numerical methods are used to solve (5) and (6).
3. Approximate Maximum Likelihood Estimation (AMLE) It is noted that from (5) and (6) the MLE of and cannot be obtained in explicit form, we derive the AMLEs which have explicit forms. Let Z ln X , then Z
extreme value distribution with pdf and cdf given respectively by
g (x )
z z exp exp , z , 1
(7)
and
z G (x ) 1 exp exp ,
(8)
where ln / and 1/ are the location and scale parameters respectively. From (7) and (8) and under AT-II PHCS the log likelihood function can be written as follows
where t i
zi
m
m
J
i 1
i 1
i 1
m ln( ) t i e t i R i e t i R m e t m
(9)
, i 1,..., m . To obtain the AMLEs we expand the function e t i in
Taylor series around the points i F 1 (1 q i ) ln( ln(1 q i )), where
qi 1
m
j k m j 1 R k
j m i 1
j 1 k m j 1 R k
m
m
Keeping only the first order derivatives, we have e ti
i i t i ,
where 5
, i 1,..., m .
i e (1 i ) and i e 0. i
i
Using these linear approximations, we can obtain the approximate log likelihood equations as
m J 1 m ( t ) R i ( i i t i ) R m ( m m t m ) 0 , i i i i 1 i 1
and
m J 1 m t (1 ( t )) R i t i ( i i t i ) R t m ( m m t m ) 0 (10) i i i i i 1 i 1
The solutions of equations (10) are
ˆˆ A ˆˆ B and
D D 2 4mE ˆ ˆ 2m where m
A
J
i x i R i i x i R m m x m i 1
i 1 J
m
R i
i 1
m
B
i
i 1
J
R i
i 1
i
i 1
m
i
i 1
i
R m m
,
J
i
i
R mm m
R i 1
,
i
R m m
m
J
i 1
i 1
D ( i 1)(x i A ) R i i (x i A ) R m m (x m A ), and m
J
i 1
i 1
E i (x i A ) 2 R i i (x i A ) 2 R m m (x m A ) 2 . It is noted that the AMLEs have explicit solutions and they can be used as initial values to obtain the MLEs in (5) and (6).
6
4. Bayesian Estimation In this section, the SE loss function is used to obtain the Bayes estimates of the unknown parameters and . The Bayes estimates is considered under the assumption of independent gamma priors of and with the following joint density g , a 1 c 1e [b d ] ,
a, b , c , d 0
(11)
Based on the likelihood function (3) and the joint prior density (11), the joint posterior distribution of , and the data is
g , y m a 1
m x i b d m c 1 i 1
e
m
J
i 1
i 1
(1 e x m )R m x i 1 (1 e x i ) Ri
(12)
under SE loss function, the Bayes estimate of the function of the parameters and , sayU , is
0
0
U ( , )
U ( , ) L ( , ) g ( , ) d d
0
0
L ( , ) g ( , ) d d
0
0
U ( , ) e Q ( , ) .d .d
0
0
e Q ( , )d .d
(13)
where Q ( , ) ln L ( , ) ln g ( , ) (a,b ) . It is not possible to compute the ratio of the two integrals given by (13) in a closed form. Therefore, in such situation, we suggest using Lindley's approximation to obtain the Bayes estimates of the unknown parameters. For the two parameter case, ( , ) (1 , 2 ) , Lindley's approximation of (13) take the form 1 1 1 u ij ij U j j L30 11U 1 L 21 2 12U 1 11U 2 2 2 2 1 1 L12 22U 1 2 12U 2 L03 22U 2 , 2 2
U ( ) U ( )
(14)
where L
2U ( ) L ( ) U ( ) 3 and , , for i , j 1,2, u , , 0,1, 2,3 u ij i i j i 1 2
U k u i ki , j i
( ) , j 1,2 and j
ij is the (i,j)th element of the inverse of the
Fisher information matrix. Evaluation all functions in equation (14) at the MLE of
7
(1 , 2 ) , produces the approximation U ( ) to (13). Now, to apply Lindley's
approximation (14), we obtain the quantities
2 m J ˆ ˆ L20 2 2 R i x i2 i u i2 R m x m2 mu m2 , ˆ i 1 L02
J 2 m ˆ m ˆ 2 ˆ R x ˆ ln(x )2u 1 ˆx ˆ u x ln( x ) i i i i i i i i i 2 ˆ 2 i 1 i 1
ˆ ˆ ˆR m x m ln(x m )2u m 1 ˆx m mu m ,
L11
m J 2 ˆ ˆ ˆ x i ln(x i ) R i ˆx i2 ln(x i )i u i x iu i ln(x i ) i 1 i 1 ˆ ˆ R m ˆx m2 ln(x m )mu m x m u m ln(x m ) ,
L30
3 2m J ˆ ˆ R i x i3 i u i2 1 2i u i R m x m3 mu m2 1 2mu m , 3 3 ˆ i 1
L 03
J 3 2m ˆ m ˆ 3 ˆ R ln(x ) 2 x ˆu ln(x ) 1 ˆx ˆ u x ln( x ) i i i i i i i i i i 3 ˆ 3 i 1 i 1
ˆ ˆ ˆ ˆx i2 i u i2 ln(x i ) 2 ˆx i 2ˆx i i u i ˆR m ln(x m ) 2 ˆ ˆ ˆ x m u m ln(x m ) 1 ˆx m m u m ˆx m2 m u m2 ln(x m ) ˆ ˆ 2 ˆx m 2ˆx m m u m ,
L 21
2ˆx u 2ˆx u
J 3 ˆ R i x i2 ln(x i )i u i2 2 i 1
ˆ R m x m2 ln(x m )m u m2
ˆ
i
i
i
ˆ m
m
m
1 x
ˆ ˆ ˆx i 1 x i2 ln(x i )i u i2 ˆ ˆx m
2 ˆ m
ln(x m )m u m2 ,
and
m J 3 ˆ ˆ ˆ ˆ L12 2 x i ln(x i ) 2 R i ln(x i ) 2 x i ln(x i )u i 1 ˆx i i u i ˆx i2 i u i2 i 1 i 1
ˆ ˆ ˆ ˆ ˆ 2 ˆx i 2ˆx i i u i R m ln(x m ) 2 x m ln(x m )u m 1 ˆx m mu m ˆx m2 i u m2 ˆ ˆ 2 ˆx m 2ˆx m m u m ,
where i e as
x i
11
. The elements of the variance covariance matrix ij , can be obtained
L02 L20 L02 L112
, 22
L 20 L 20 L02 L112
Based on the joint prior function (11), we obtain
8
and 12 21
L11 L 20 L02 L112
.
( , ) ln g ( , ) (a 1)ln (c 1)ln b d hence 1
( , ) a 1 ( , ) c 1 b and 2 d .
If U ( , ) , u 1 1, u 2 0 and u11 u12 u 21 u 22 0 , then ˆ 1 11 2 12
L30 112 3L 21 11 12 L12 ( 11 22 2 122 ) L03 12 22 2
Similarly, when U ( , ) , hence, u 2 1, u 1 0 and u11 u12 u 21 u 22 0 . Thus ˆ 1 12 2 22
2 L30 11 12 L 21 2 122 11 22 3L12 22 12 L03 22
2
5. Data Analysis and Simulation Results The purpose of this section is to compare the performance of the different methods of estimation discussed in the previous sections. We analyze a real data set for illustrative purpose; also, a simulation study is carried out to check the behavior of the proposed methods as well as to evaluate the statistical performances of the estimators under different sampling schemes. 5.1. Real Data Example In this subsection, we reanalyze a real data set given by Blischke and Murthy (2000). The data set consists of 87 observations and it represents the failure times of aircraft windshields. Blischke and Murthy (2000) proved that the IW distribution gives a good fit for it. From the original data we generate three adaptive progressively hybrid censored samples with the following schemes Scheme
m
1
40
T 2.5
2
50
3
3
70
3.5
Censoring scheme
0*20,1*19, 28 1*25,0*24,12 17,0*69
the generated adaptive progressively hybrid censored samples given in the following table
9
Scheme 1
2
3
0.301
0.309
0.557
0.943
Censored data 1.070 1.124 1.248
1.281
1.281
1.303
1.480
1.505
1.506
1.568
1.615
1.619
1.652
1.652
1.757
1.795
1.866
1.899
1.911
1.981
2.010
2.085
2.089
2.097
2.154
2.194
2.223
2.300
2.349
2.481
2.625
2.632
2.661
2.823
2.902
2.934
0.301
0.309
0.557
0.943
1.124
1.248
1.281
1.281
1.432
1.480
1.505
1.568
1.615
1.652
1.652
1.795
1.866
1.876
1.899
1.911
1.912
1.981
2.038
2.085
2.135
2.154
2.194
2.224
2.229
2.324
2.385
2.610
2.632
2.646
2.688
2.890
2.934
2.962
3.000
3.114
3.117
3.344
3.376
3.385
3.467
3.478
3.578
3.595
3.699
3.779
0.301
0.309
0.557
0.943
1.070
1.248
1.281
1.281
1.432
1.480
1.505
1.506
1.615
1.619
1.652
1.652
1.757
1.795
1.866
1.876
1.899
1.911
1.912
1.914
1.981
2.038
2.085
2.089
2.097
2.135
2.154
2.190
2.194
2.223
2.229
2.324
2.349
2.481
2.610
2.625
2.632
2.646
2.661
2.688
2.823
2.890
2.934
2.962
3.000
3.103
3.117
3.344
3.376
3.385
3.467
3.478
3.578
3.595
3.779
4.035
4.121
4.240
4.255
4.278
4.305
4.449
4.485
4.570
4.602
4.694
The MLEs, AMLEs and Bayes estimates using Lindley's approximation are reported in table 1. The Bayes estimates of and are computed based on a non-informative prior, i.e., a b c d 0 . From table 1, it is noted that the MLEs and Bayes estimates of the unknown parameters are quite close to each other. Also, the estimates based on adaptive progressively hybrid type-II samples are very close to those of complete data and this demonstrate the importance and usefulness of AT-II PHCS. Table 1: MLEs, AMLEs and Bayes estimates of and . Scheme Complete 1 2 3
2.157 2.335 2.288 1.984
MLEs
AMLEs
1.392 0.965 1.128 1.353
282.762 68.277 33.14 102.357
4.277 4.086 2.799 3.550
Bayes estimates
2.149 2.325 2.279 1.975
1.387 0.957 1.121 1.348
5.2. Simulation Study This subsection aims to conduct a Mont Carlo simulation study to compare the performance of the different methods of estimation proposed in the previous sections. The performance of all the estimates is compared in terms of their mean square errors (MSEs). The average values and MSEs of MLEs, AMLEs and Bayes estimates of and are evaluated for , 0.5,1 and , 0.75, 2 and by considering different values n , m , T and three censoring schemes as follows 10
Scheme 1: R1 · · · R m 1 0 and R m n m , Scheme 2: R1 · · · R m 1 1 and R m n 2m 1 , and Scheme 3: R1 · · · R m 1 = R m (n m ) / m . We replicate the process 1000 times. The average values and MSEs are tabulated in tables 3-6 . The Bayes estimates are all computed under non-informative and informative priors. The priors for the two sets of parameters are given in table 2. We use prior 0 for the two sets of parameter values, Prior 1 for , 0.5,1 and prior 2 for , 0.75, 2 . From table 2, it may be noted that prior 1 describes the case of non-informative prior and prior 1 and prior 2 are chosen in such way that prior means equal to the original means. Table 2: Different priors for the two sets of parameters of and . Parameters Prior ( 0.5, 1) and ( 0.75, 2) ( 0.5, 1) ( 0.75, 2)
Prior 0: a b c d 0. Prior 1: a 1, b , 2, c 1, d 1. Prior 2: a 3, b , 4, c 4, d 2.
From tables 3-6, it can be seen that the Bayes estimates using prior 1 perform better than those by using AMLEs, MLEs and Bayes estimates using prior 0 in terms of minimum MSEs for , 0.5,1 , while the AMLEs has the minimum MSEs among all other methods for , 0.75, 2 . It is observed that the Bayes estimates with respect to the non-informative prior (prior 0) are quite close to the MLEs. For fixed n and T , when m increases the MSEs of MLEs, AMLEs and Bayes estimates decrease except come cases. For fixed n and m , when T increases we do not observe any specific trend in the MSEs. In most cases, for the parameter , the MSEs of scheme 3 are always smaller than other schemes, while, for the parameter the scheme 1 has the minimum MSEs, except some cases.
11
Table 3: The average values and MSEs of the parameters , 0.5,1 under different censoring schemes and for T 0.5 . (n , m )
Scheme
Parameter
AML Estimates
ML Estimates
(30,10)
1
0.84120 0.12074 1.38931 0.16330
0.62652 0.08001 1.01470 0.10278
Prior 0 0.67608 0.09109 1.01639 0.09627
Prior 1 0.63785 0.06194 1.00692 0.06646
0.80237 0.09563 1.43781 32891.0
0.55475 0.08011 1.05662 0.11566
0.59505 0.08183 1.02897 0.10672
0.57089 0.05796 1.03481 0.07282
0.76593 0.07328 1.48584 0.24096
0.50452 0.07075 1.12891 0.15532
0.53115 0.06965 1.10470 0.14096
0.51538 0.04991 1.09623 0.09344
0.89938 0.16010 1.46719 0.22143
0.72228 0.12108 1.03026 0.08340
0.79207 0.15587 1.06119 0.08097
0.72570 0.09724 1.01944 0.05013
0.88584 0.14959 1.49804 0.25100
0.68010 0.09758 1.01656 0.08649
0.74499 0.12429 1.04696 0.08307
0.38939 0.07874 1.01312 0.05179
0.83352 0.11231 1.56630 0.32316
0.55118 0.06373 1.18009 0.18966
0.50708 0.06171 1.15314 0.17083
0.51742 0.04452 1.12570 0.09651
0.87597 0.14202 1.45344 0.20808
0.74573 0.10340 1.08966 0.05249
0.78807 0.12475 1.11037 0.05396
0.74330 0.09172 1.07632 0.04199
0.85861 0.12949 1.49069 0.24306
0.65411 0.06174 1.04743 0.04983
0.69224 0.07405 1.06676 0.05005
0.66041 0.05486 1.04078 0.03940
0.80155 0.09195 1.54896 0.30328
0.50084 0.04940 1.07197 0.09662
0.52877 0.04966 1.05670 0.09179
0.51678 0.03961 1.05788 0.06853
0.83662 0.11570 1.41143 0.17351
0.66112 0.05742 1.07117 0.04028
0.68658 0.06507 1.08558 0.04134
0.66317 0.05247 1.06426 0.03535
0.79318 0.08829 1.45456 0.20926
0.57892 0.05319 1.02938 0.05373
0.59918 0.05519 1.04213 0.05295
0.58438 0.04592 1.02974 0.04678
0.75834 0.06810 1.50145 0.25351
0.52652 0.04541 1.02493 0.05805
0.54495 0.04580 1.01455 0.05638
0.53430 0.03829 1.02196 0.04933
2
3
(50,10)
1
2
3
(60,15)
1
2
3
(60,20)
1
2
3
12
Bayes Estimates
Table 4: The average values and MSEs of the parameters , 0.5,1 under different censoring schemes and for T 1 . (n , m )
Scheme
Parameter
AML Estimates
ML Estimates
(30,10)
1
0.84926 0.12332 1.39117 0.15957
0.73272 0.11794 1.04086 0.08473
Prior 0 0.78493 0.14242 1.07073 0.08296
Prior 1 0.72916 0.09645 1.02981 0.05671
0.82409 0.10653 1.44180 0.20065
0.60422 0.05799 1.02692 0.08288
0.65013 0.06810 1.00187 0.07749
0.61638 0.04571 1.02223 0.05450
0.78353 0.08306 1.48678 0.24150
0.51580 0.06699 1.31944 0.32484
0.58024 0.06029 1.29266 0.29855
0.58026 0.04532 1.23237 0.16178
0.89989 0.16050 1.46482 0.21908
0.71648 0.11728 1.02301 0.07936
0.78650 0.15136 1.05425 0.07642
0.72134 0.09437 1.01192 0.04864
0.88556 0.14932 1.49425 0.24722
0.69129 0.10087 1.01951 0.08411
0.75653 0.12912 1.04985 0.08086
0.69904 0.08178 1.01299 0.04973
0.84124 0.11734 1.56069 0.31697
0.63707 0.05350 1.28768 0.22532
0.59261 0.04442 1.25882 0.20335
0.58789 0.03295 1.21168 0.11488
0.87473 0.14104 1.44579 0.20162
0.77594 0.12545 1.09747 0.05273
0.81770 0.14904 1.11672 0.05500
0.76998 0.11080 1.07996 0.04232
0.85548 0.12704 1.48228 0.23530
0.72880 0.09645 1.08661 0.05005
0.76672 0.11421 1.10513 0.05186
0.72682 0.08560 1.07331 0.04064
0.81379 0.09948 1.54321 0.29723
0.60824 0.04596 1.22298 0.15309
0.58022 0.04095 1.20485 0.14204
0.58037 0.03381 1.18671 0.10617
0.83568 0.11345 1.40612 0.16856
0.81355 0.13249 1.14773 0.04968
0.83929 0.14835 1.16119 0.05294
0.80397 0.12113 1.13289 0.04320
0.80755 0.09546 1.45627 0.21124
0.69775 0.06476 1.10287 0.03802
0.72027 0.07353 1.11557 0.03999
0.69518 0.05949 1.09319 0.03346
0.76361 0.07080 1.50307 0.25555
0.54139 0.03354 1.11271 0.08441
0.52242 0.03203 1.10111 0.08013
0.52791 0.02736 1.10137 0.06772
2
3
(50,10)
1
2
3
(60,15)
1
2
3
(60,20)
1
2
3
13
Bayes Estimates
Table 5: The average values and MSEs of the parameters , 0.75, 2 under different censoring schemes and for T 1 . (n , m )
Scheme
Parameter
AML Estimates
ML estimates
(30,10)
1
0.80937 0.00670 2.40940 0.27804
0.97243 0.12133 2.02838 0.32448
Prior 0 1.01723 0.13846 2.08998 0.31099
Prior 2 0.96578 0.07229 2.09384 0.09764
0.77352 0.00463 2.56589 0.39789
0.82462 0.06817 2.10313 0.37125
0.86329 0.07100 2.04529 0.34155
0.85689 0.03292 2.05653 0.09090
0.77801 0.00422 2.70167 0.54953
0.81926 0.09084 2.40655 0.92012
0.78693 0.08408 2.35887 0.85149
0.75430 0.03356 2.02725 0.19757
0.87252 0.01585 2.63341 0.43911
1.05224 0.18545 2.08213 0.34465
1.11549 0.22284 2.14326 0.33693
1.02722 0.10679 2.10348 0.09149
0.85626 0.01219 2.72912 0.56563
1.02066 0.16279 2.08233 0.32825
1.07942 0.19344 2.14169 0.32144
1.00374 0.09451 2.11278 0.09292
0.80559 0.00435 2.94120 0.91059
0.86882 0.08934 2.44654 0.79588
0.82355 0.07901 2.38578 0.72915
0.75794 0.03493 2.01157 0.78184
0.84135 0.00912 2.56649 0.35271
1.07890 0.17779 2.19988 0.20610
1.13491 0.20186 2.23831 0.21579
1.06367 0.12897 2.15770 0.11154
0.81974 0.00568 2.68623 0.49808
1.04386 0.13741 2.18700 0.19983
1.07635 0.15549 2.22380 0.20813
1.01778 0.09975 2.16041 0.11244
0.76626 0.00144 2.88341 0.79959
0.79554 0.06345 2.22991 0.39298
0.76933 0.06002 2.19805 0.36808
0.75069 0.03192 2.07146 0.12119
0.79880 0.00368 2.46771 0.25553
1.05816 0.13076 2.22537 0.15954
1.07993 0.14337 2.25353 0.16944
1.03358 0.10414 2.19499 0.11358
0.76764 0.00222 2.62354 0.41512
0.86429 0.04151 2.05494 0.13237
0.88375 0.04522 2.08244 0.13210
0.86827 0.03185 2.07294 0.08740
0.78881 0.00281 2.74665 0.57809
0.77950 0.03809 2.15901 0.21259
0.76286 0.03636 2.13672 0.20164
0.75235 0.02327 2.08142 0.10755
2
3
(50,10)
1
2
3
(60,15)
1
2
3
(60,20)
1
2
3
14
Bayes Estimates
Table 6: The average values and MSEs of the parameters , 0.75, 2 under different censoring schemes and for T 1.5 . (n , m )
Scheme
Parameter
AML Estimates
ML estimates
(30,10)
1
0.80063 0.00484 2.35464 0.24384
1.08155 0.19673 2.11599 0.35206
Prior 0 1.12358 0.22080 2.17462 0.34848
Prior 2 1.04211 0.12138 2.13630 0.11575
0.76925 0.00269 2.52477 0.36646
0.95569 0.11513 2.02937 0.32898
0.99240 0.12717 2.08430 0.31716
0.94513 0.06750 2.09950 0.10404
0.76182 0.00339 2.69557 0.53999
0.92067 0.10518 2.67367 1.29705
0.88585 0.09175 2.61847 1.19010
0.80097 0.03047 2.04240 0.18736
0.87179 0.01571 2.62097 0.42686
1.06107 0.19045 2.07749 0.34702
1.12404 0.22894 2.13861 0.33905
1.03381 0.11103 2.10085 0.09573
0.80555 0.01207 2.72037 0.55510
1.02722 0.16575 2.07678 0.33217
1.08566 0.19734 2.13629 0.32462
1.00775 0.09785 2.10681 0.11298
0.80381 0.00396 2.93084 0.89270
0.86408 0.06718 2.42988 0.65334
0.81913 0.05780 2.37646 0.59091
0.75530 0.01844 2.04042 0.06849
0.84097 0.00917 2.56394 0.35049
1.09847 0.18361 2.19213 0.20527
1.08445 0.20742 2.23079 0.21443
1.06455 0.13316 2.15411 0.11037
0.84087 0.00914 2.56455 0.35102
1.10032 0.18444 2.19536 0.20795
1.09626 0.20836 2.23396 0.21721
1.06585 0.13353 2.15630 0.11316
0.77329 0.00159 2.87405 0.78316
0.86897 0.05540 2.38978 0.45160
0.84140 0.04869 2.35430 0.41630
0.80016 0.02330 2.17473 0.12152
0.79119 0.00280 2.41656 0.21571
1.08908 0.18348 2.27255 0.18525
1.14962 0.19805 2.29980 0.19727
1.09439 0.14591 2.23131 0.13355
0.75871 0.00121 2.58340 0.37295
1.01324 0.10422 2.21404 0.16030
1.03098 0.11268 2.23927 0.16840
0.99195 0.08199 2.19336 0.11776
0.78150 0.00246 2.74098 0.57189
0.83505 0.04505 2.30239 0.34530
0.81766 0.04146 2.27811 0.32513
0.79505 0.02512 2.17909 0.15389
2
3
(50,10)
1
2
3
(60,15)
1
2
3
(60,20)
1
2
3
15
Bayes Estimates
6. Optimal Censoring Scheme In practice, it is very important to choose the optimum censoring scheme to obtain the highest amount of information about the unknown parameters. In the case of AT-II PHCS, for fixed n and m , the practitioner might be interested to choose the ideal time T and the progressive censoring scheme R1 ,..., R m , where n m i 1 R i m
and R J 1 , R J 2 ,..., R m 1 0 if X J :m :n T X J 1:m :n , which provides more information about the unknown parameters under consideration. Here, we investigate the optimal censoring schemes in terms of (a) Criteria 1: minimum trace of the variancecovariance matrix of the MLEs ( ) and (b) Criteria 2: minimizing the variance of the estimate of the p-th quantile given by
V (Qˆ p ) V 11 (ˆ) F( p1) V 22 (ˆ) 2F(11p )V 12 (ˆ) 2
where Qˆ p is the MLE of the p-th quintile and F( p1) is the inverse CDF of the IW distribution defined as 1 (p)
F
ln( p )
1/
Illustrative Example: Using the real data set of Blischke and Murthy (2000), We generate an adaptive progressively hybrid censored samples considering different values of m , T and censoring schemes. The comparison results of different optimal censoring schemes are presented in tables 7-10 by obtaining and the variance of 5-th quantiles V (Qˆ0.05 ) and 95-th quantiles V (Qˆ0.95 ) .
Table 7: Different censoring planes for m 30 and T 2 . Scheme
ˆ
ˆ
0*29,57
2.3080
0*20,1*9, 48 1*29, 28 2*10,1*10,0*9, 27 1*20, 2*9,19 57,0*29 0*10,1*10,0*9, 47
0.9320
V (Qˆ0.05 ) 0.6870
V (Qˆ0.95 ) 48.3870
0.0820
2.3170
0.9220
0.7404
52.4120
0.0821
2.4009
0.8718
1.1089
80.881
0.0894
2.3683
0.8963
0.8726
63.136
0.0896
2.3599
0.9122
0.8193
56.501
0.0868
1.3827
1.2417
0.2051
4.2986
0.0806
2.3654
0.9044
0.8865
61.6079
0.0851
Table 8: Different censoring planes for m 30 and T 2.5 . 16
Scheme
ˆ
ˆ
0*29,57
2.3080
0*20,1*9, 48 1*29, 28 2*10,1*10,0*9, 27 1*20, 2*9,19 57,0*29 0*10,1*10,0*9, 47
0.9320
V (Qˆ0.05 ) 0.6870
V (Qˆ0.95 ) 48.3870
0.0820
2.3890
0.9020
0.9680
64.0130
0.0870
2.4390
0.8490
1.3820
101.8880
0.0920
2.5140
0.8120
1.8590
148.9720
0.1000
2.4920
0.8150
1.8820
147.4520
0.0960
1.9755
1.3453
0.2105
5.3845
0.1522
2.3450
0.9140
0.8060
56.4180
0.0840
V (Qˆ0.95 ) 15.1367
0.0739
Table 9: Different censoring planes for m 40 and T 2 . Scheme
ˆ
ˆ
0*39, 47
2.2342
1.0629
V (Qˆ0.05 ) 0.3977
0*20,1*19, 28 1*39,8
2.1421
1.1307
0.2224
9.8835
0.0714
2.3354
1.018
0.5452
20.4027
0.0821
2*15,1*10,0*14,7 1*9, 2*19,0*12 47,0*39 0*15,1*15,0*9,32
2.5324
0.9137
1.1716
46.9090
0.0958
2.4130
1.0077
0.6174
22.4141
0.0873
1.5517
1.2985
0.0443
3.3231
0.0759
2.2898
1.0057
0.5575
21.9797
0.0770
Table 10: Different censoring planes for m 40 and T 2.5 . Scheme
ˆ
ˆ
0*39, 47
2.2342
0*20,1*19, 28 1*39,8 2*15,1*10,0*14,7 1*9, 2*19,0*12 47,0*39 0*15,1*15,0*9,32
1.0629
V (Qˆ0.05 ) 0.3977
V (Qˆ0.95 ) 15.1367
0.0739
2.2612
1.0248
0.4949
19.2707
0.0753
2.2750
1.0316
0.4437
18.1368
0.0785
2.3690
1.0263
0.4804
19.2229
0.0870
2.3535
1.0211
0.4702
19.9133
0.0836
1.6393
1.3048
0.0068
3.4906
0.0827
2.2893
0.9886
0.6055
24.6367
0.0769
From tables 7-10, it is noted that the censoring scheme number six has the minimum variance of the estimate of the p-th quantile, i.e. V (Qˆ0.05 ) and V (Qˆ0.95 ) .
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7. Conclusion In this paper, we discussed the estimation problem of the unknown parameters of the IW distribution based on adaptive type-II progressively hybrid censored data. We used classical and Bayesian estimation methods to estimate the unknown parameters. It is observe that the MLEs cannot be obtained in explicit forms, therefore, the AMLEs have been suggested. We obtained the Bayes estimates based on SE loss function under the assumption of independent gamma priors using Lindley's approximation. A real data set is used to show how the scheme works in practice. The performance of the different estimators is compared based on simulation study in terms of their MSEs. It is observed that the Bayes estimates with non-informative priors work very well in terms of MSEs, also, the Bayes estimates using the informative prior have the smallest MSEs among all other estimates proposed. Finally, we have proposed the optimal censoring scheme based on two different criteria and an illustrative example is provided. As a future work, the inferential results of some life time models under adaptive type-II progressively hybrid censored data with random removals have not been developed yet. Also, one sample prediction based on AT-II PHCS remains open.
Acknowledgments The authors thank the anonymous referee for a careful reading of the article.
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