Estimation for the exponentiated Weibull model with adaptive Type-II progressive censored schemes

Accepted Manuscript

Estimation for the Exponentiated Weibull Model with Adaptive Type-II Progressive Censored Schemes Mashail M. Al-Sobhi, Ahmed A. Soliman PII: DOI: Reference:

S0307-904X(15)00405-9 10.1016/j.apm.2015.06.022 APM 10640

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

1 March 2014 19 April 2015 5 June 2015

Please cite this article as: Mashail M. Al-Sobhi, Ahmed A. Soliman, Estimation for the Exponentiated Weibull Model with Adaptive Type-II Progressive Censored Schemes, Applied Mathematical Modelling (2015), doi: 10.1016/j.apm.2015.06.022

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Highlights • Bayes and ML estimations for the exponentiated Weibull distribution based an adaptive progressive censoring have been obtained. • The MLEs, the bootstrap confidence intervals and the asymptotic confidence intervals have been obtained and discussed. • The Bayes estimates cannot be obtained in explicit form.

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• We used MCMC samples to compute the approximate Bayes estimates and constructed the credible intervals.

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• The performance of different methods was compared via a Monte Carlo simulation.

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Estimation for the Exponentiated Weibull Model with Adaptive Type-II Progressive Censored Schemes 1

Mathematics Department, Umm-Al-Qura University, Makkah, Saudi Arabia, [email protected] 2

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Mashail M. Al-Sobhi and 2 Ahmed A. Soliman

Mathematics Department, Islamic University, Madina, Saudi Arabia, a a [email protected]

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Abstract:In reliability and life testing experiments, the censoring scheme which can balance between the total time spent for the experiment, the number of units used and the efficiency of statistical inference based on the results of the experiment is desirable. An adaptiveType-II progressive censoring schemes have been shown to be useful in this case. This article deals with the problem of estimating parameters, reliability and hazard functions of the two-parameter exponentiated Weibull distribution, under adaptive progressive Type II censoring samples using Bayesian and non-Bayesian approaches. Maximum likelihood estimates (M LEs) are proposed for unknown quantities. The asymptotic normality of the MLEs are used to compute the approximate confidence intervals for these quantities, parametric bootstrap confidence intervals are also constructed. Markov Chain Monte Carlo (MCMC) samples using importance sampling scheme are used to produce the Bayes estimates and the credible intervals for the unknown quantities. A real-life data-set is analysed to illustrate the proposed methods of estimation. Finally, results from simulation studies assessing the performance of the maximum likelihood and Bayes estimators are discussed.

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Keywords: Exponentiated Weibull distribution; An adaptive Type-II progressive censoring scheme; Bayesian non-Bayesian approaches; Asymptotic confidence intervals; parametric bootstrap confidence intervals; Markov chain Monte Carlo; Importance sampling scheme.

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Introduction

The exponentiated Weibull (EW) distribution (which is denoted by EW(α, θ)) was introduced by Mudholkar and Srivastava [1]. This distribution is an extension of the well-known Weibull distribution by adding an additional shape parameter. The EW family contains distributions with nonmonotone failure rates besides a broader class of monotone failure rates. The EW distribution as a failure model is more realistic than that of monotone failure rates and plays an important role 2

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in the analysis many types of survival data. It has been well established in the literature that the EW distribution provides significantly better fits than traditional models based on the exponential, gamma, Weibull and log-normal distributions. A recent survey on the EW distribution can be found in the excellent review by Nadarajah et al. [2]. The form of the probability density function (pdf) and cumulative distribution function (cdf) of the EW distribution with two shape parameters, α and θ are given, respectively, by f (x; α, θ) = αθxα−1 exp(−xα )(1 − exp(−xα ))θ−1 , x > 0,

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and

α, θ > 0,

F (x; α, θ)= (1 − exp(−xα ))θ ,

(1) (2)

Also, the reliability and hazard functions of the EW(α, θ) distribution at mission time t are given by S(t) = 1 − (1 − exp(−tα ))θ , t > 0, (3)

h(t) =

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and

αθtα−1 exp(−tα )(1 − exp(−tα ))θ−1 , [1 − (1 − exp(−tα ))θ ]

t > 0.

(4)

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Mudholkar and Hutson [3] showed that the density function of the EW distribution is decreasing when αθ ≤ 1 and unimodal when αθ > 1. The hazard function allows for constant, monotonically increasing, monotonically decreasing, unimodal and bathtub shaped hazard rates. In particular, bathtub shapes when α > 1 and αθ < 1. Unimodal shapes with α < 1 and αθ > 1. Monotonically increasing shapes occur when α > 1 and αθ > 1. Monotonically decreasing shapes occur when α < 1 and αθ < 1. The hazard rate is constant when α = θ = 1. The applications of the EW distribution have been widespread. For examples, its used for modeling of: extreme value data using floods, tree diameters, firmware system failure, the survival pattern of test subjects after a treatment is administered to them, distribution for excess-of-loss insurance data, software reliability data, bus-motor failure data, mean residual life computation of (n - k + 1)-out-of-n systems and other models. For more details of these applications see [2]. Maximum likelihood estimations (besides testing of hypotheses) for the EW distribution using several sets of data are discussed by Mudholkar et al.[4]. Parametric characterizations of the density function are discussed in [3] and [5]. Other statistical properties of this distribution are discussed in [6]. Nassar and Eissa [7] derived Bayes estimates of the two shape parameters, reliability and failure rate functions of the EW distribution from complete and Type II censored samples. Pal et al.[8] introduced many properties and obtained some inferences for the three parameter EW distribution. Kim et al. [9] obtained the maximum likelihood and Bayes estimators for the two shape parameters and the reliability function of the EW model based on progressive Type-II censored samples. Some Bayesian inferences based on generalized order statistics from the EW distributions using Markov chain Monte Carlo (MCMC) methods are discussed in [10]. For recent ar 3

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In life testing and reliability experiments, reducing the total test time and the associated cost is one of the major reason. This reason leads us to the area of censoring schemes. A censoring scheme can be balance between total time spent for the experiment; number of units used in the experiment; and the efficiency of statistical inference based on the results of the experiment. There are several types of censored schemes. The most common censoring schemes are Type-I (time) censoring, and Type-II (failure) censoring. However, the conventional Type-I and Type-II censoring schemes do not have the flexibility of allowing removal of units at points other than the terminal point of the experiment. Because of this lack of flexibility, a more general censoring schemes are introduced. The progressive Type-II right censoring scheme is an appealing one and has attracted much attention in the literature. A good account on progressive censoring schemes can be found in the monograph [11], or in the excellent review by Balakrishnan [12]. A crucial assumption in the design of the progressively censored experiment is that the censoring scheme (the numbers of withdrawing units) is known in advance. However, although this assumption is normally assumed in the literature, it may not be satisfied in real-life experiments since the experimenter may change the censoring numbers during the experiment (for any reasons). Therefore, it is desirable to have a model that takes into account such an adaption process. Such a model is proposed by Ng et al. [13] and called an adaptive Type-II progressive censoring scheme. For this censoring scheme, the effective sample size is fixed in advance, and the progressive censoring scheme is provided but the number of items progressively removed from the experiment upon failure may change during the experiment. In this article, we suggest an adaptive Type-II progressive censoring scheme, this scheme can be save both the total test time and the cost induced by failure of the units and increases the efficiency of statistical analysis. The scheme of adaptive progressive censoring has been discussed for some distribution (see, e.g., Balakrishnan and Cramer [14]. The rest of the article is organized as follows. In Section 2, we describe the adaptive Type-II progressive censoring scheme. The maximum likelihood estimation is considered in Section 3. In Section 4, asymptotic confidence intervals are constructed for unknown quantities using the normality property of corresponding MLEs. Two parametric bootstrap confidence intervals are obtained in Section 5. In Section 6, we discuss importance sampling scheme to compute the Bayes estimates. Markov Chain Monte Carlo (MCMC) samples are further utilized to construct credible intervals for unknown quantities. A real-life data-set is analyzed in Section 7 to illustrate the proposed methods of estimation. A numerical study has been made in Section 8 for comparison purposes using monte Carlo simulations. Confidence intervals obtained by different methods are also compared in terms of their expected widths and coverage probabilities. Finally, we conclude the paper in Section 9.

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An adaptive Type-II progressive censoring scheme

Type-II censoring scheme is often used in life testing experiment. Only m units in a random sample of size n (m < n) are observed. Progressive Type-II censoring is a generalization of Type-II 4

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censoring. The progressive Type-II censoring scheme can be briefly described as: Suppose n units are placed on a life testing experiment and let X1 , X2 , ..., Xn are their corresponding lifetimes. Let Xi , i = 1, 2, ..., n are independent and distributed with probability density function (pdf ) f (x) and cumulative distribution function (cdf ) F (x). Prior to the experiment, an integer m < n is determined and the progressive Type-II censoring scheme R = (R1 , R2 , ..., Rm ) with Ri ≥ 0 and Pm i=1 Ri +m = n is specified. At the time of occurrence of the i−th failure, the Ri of the surviving units are withdrawn randomly from the experiment. The experiment continues until the m failures are observed. We denote the m completely observed lifetimes by Xi:R m: n , i = 1, 2, ..., m, which are the observed progressively Type-II right censored order statistics. A crucial assumption in the design of the progressively censored experiment is that the censoring scheme (R1 , R2 , ..., Rm ) is known in advance, i.e. the integers R1 , R2 , ..., Rm are prefixed. From an experimental design point of view, this approach may be interesting in the sense of finding an initially optimal scheme, see for example [15]. However, this assumption may not be satisfied in real-life experiments since the experimenter may change the censoring numbers during the experiment (for some reasons). Kundu and Joarder [16] proposed a censoring scheme called Type- II progressive hybrid censoring scheme, in which a life testing experiment with progressive Type-II right censoring scheme (R1 , R2 , ..., Rm ) is terminated at a prefixed time T . However, the drawback of the Type-II progressive hybrid censoring, similar to the conventional Type-I censoring (time censoring), is that the effective sample size is random and it can turn out to be a very small number (even equal to zero) and therefore the standard statistical inference procedures may not be applicable or they will have low efficiency. Ng et al. [13] suggest an adaptive Type-II progressive censoring, in this censoring scheme, a properly planned adaptive progressively censored life testing experiment can save both the total test time and the cost induced by failure of the units and increase the efficiency of statistical analysis. In the adaptive type-II progressive censoring scheme, the effective sample size m is fixed in advance and the progressive censoring scheme (R1 , R2 , ..., Rm ) is provided, but the values of some of the Ri may change accordingly during the experiment. The adaptive typeII progressive censoring scheme works as follows: Suppose the experimenter provides a time T , which is an ideal total test time, but we may allow the experiment to run over time T . If Xm:m:n < T , the experiment proceeds with the pre-specified progressive censoring scheme (R1 , R2 , ..., Rm ) and stops at the time Xm:m:n (see Figure (1)). Otherwise, once the experimental time passes time T but the number of observed failures has not reached m, we would want to terminate the experiment as soon as possible for fixed value of m, then we should leave as many surviving items on the test as possible. Suppose J is the number of failures observed before time T , i.e. XJ:m:n < T < XJ+1:m:n , J = 0, 1, ..., m,

where X0:m:n ≡ 0 and Xm+1:m:n ≡ ∞. After passed time T , we do not withdraw any items at all except for the time of the m − th failure where all remaining surviving items are removed. P Therefore, we set RJ+1 = ... = RJ−1 = 0 and Rm = n − m − ji=1 Ri , i.e., the effectively applied 5

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R1 , ..., Rj ∗ , 0, 0, 0, n − m −

j X

Ri

i=1

!

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R1

R2

x1:m:n

x 2:m:n

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The basic idea of this scheme is to speed up the test as much as possible when the test duration exceeds a pre-determined threshold T . Its illustrates how an experimenter can control the experiment. If he is interested in getting observations early, he will remove less units (or even none). If he wants to have larger observed failure times, he will remove more units. Figure (2) gives the schematic representation of this situation. The value of T plays an important role in the determination of the values of Ri and also as a compromise between a shorter experimental time and a higher chance to observe extreme failures. When T = ∞, the adaptive variant reduces to a progressive Type- II censoring one with censoring scheme (R1 , ..., Rm ). If T = 0, this adaptive variant leads to a conventional Type-II censoring scheme.

R m −1

…

x m −1:m:n

Rm

x m:m:n

Figure (1) Experiment terminates before time T (i.e.

R1

R2 x 2:m:n

Withdrawn

Withdrawn J

n − m − ∑ Ri

RJ

…

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x1:m:n

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x J :m:n

i =1

T

x J +1:m:n

… x m:m:n End

x m:m:n ≥ T )

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Figure (2) Experiment terminates after time T (i.e.

Maximum Likelihood Estimation

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x m:m:n < T )

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R Suppose that X = Xi;m,n , i = 1, 2, ..., m, be an adaptive Type-II progressive censored sample of size m from a sample of size n taken from distribution having probability density function (P DF ) f (x) and cumulative distribution function (CDF ) F (x), with the associated progressive censoring scheme R = (R1 , R2 , ..., Rm ). Given J = j, the corresponding likelihood function based on this data is given by

R R f (xR 1;m,n , x2;m,n , ..., xm;m,n ) = BJ

"

m Y i=1

×[1 −

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#

"

J Y R Ri f (xi;m,n ) × [1 − F (xR i;m,n )]

Cj F (xR m;m,n )] ,

i=1

# (5)

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where

min{i−1,J} m Y X BJ = [n − i + 1 − Rk ] i=1

and Cj = n − m −

k=1

J X

Ri

i=1

In this section, we consider maximum likelihood estimation of unknown two parameters, the reliability function S(t) and the hazard function h(t). Using Equations (1) and (2) in Equation (5), the likelihood function of α and θ is obtained as m m

L (α, θ | x) ∝ α θ where

m Y

Vi Uiθ−1

i=1

#

×

" J Y i=1

i (ζ R i )

The log-likelihood function may then be written as

` (α, θ | x) = log L (α, θ | x) ∝ m log α + m log θ + +

× [ζ m ]Cj ,

        

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xi ≡ xR i;m,n , Ui (α) ≡ Ui = 1 − exp(−xαi ) Vi (α) ≡ Vi = xα−1 exp(−xαi ) i ζ i (α, θ) ≡ ζ i = 1 − Uiθ

J X

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m X i=1

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(7)

((α − 1) log xi − xαi + (θ − 1) log Ui )

(Ri log ζ i ) + Cj log ζ m .

i=1

(6)

(8)

Next, we observe that the likelihood equations of α and θ are m

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X ∂ m X ` (α, θ | x) = + (η 1i + (θ − 1)η 2i ) − (Ri φi ) − Cj φm = 0, ∂α α i=1 i=1

and

m

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X ∂ m X ` (α, θ | x) = + log Ui − (Ri Qi ) − Cj Qm = 0. ∂θ θ i=1 i=1

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where η 1i = (1 − xαi ) log xi , $i = xαi exp[−xαi ] log xi ,

Qi =

(9)

η 2i =

$i , Wi Ui

Uiθ log Ui . ζi

=

$i , ζi

(10)

φi = θUiθ−1 Wi , and

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The MLEs of α and θ, respectively, α ˆ and ˆθ , are the solution of the two equations (9) and (10), which cannot be solved analytically. So, we have used a numerical method such as Newton, s method to determine the desired estimates. Further, we obtained the MLEs of S(t) and h(t), ˆ ˆ and h(t), respectively, S(t) using the invariance property of the MLEs, as ˆ = 1 − (1 − exp(−tαˆ ))ˆθ , t > 0, S(t)

(11)

ˆ ˆ ˆ =α h(t) ˆ ˆθtαˆ −1 exp(−tαˆ )(1 − exp(−tαˆ ))θ−1 [1 − (1 − exp(−tαˆ ))θ ]−1 , t > 0.

(12)

and

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4

Asymptotic confidence intervals

confidence intervals for α and θ From the log-likelihood function in (7), we have m

m

X ∂ 2 ` (α, θ | x) m X α 2 = − 2− (xi log xi ) − (θ − 1) η 2i (η 2i − η 1i ) 2 ∂α α i=1 i=1

j X (Ri φi ( (θ − 1) η 2i + η 1i + φi ) − Cj φm ( (θ − 1) η 2m + η 1m + φm ), (13) i=1

m

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−

X ∂ 2 ` (α, θ) X ∂ 2 ` (α, θ | x) = = η 2i − (Ri φi ∆i ) − Cj φm ∆m , ∂α∂θ ∂θ∂α i=1 i=1

and

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∂ 2 ` (α, θ | x) −m X log2 Ui log2 Um ) − C Q , = − (R Q j m i i ζ ζ ∂θ2 θ2 i m i=1

(14)

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where ∆i = ( 1θ + log Ui + Qi ). The asymptotic variance–covariances of the MLE for parameters α and θ are given by the elements of the inverse of the Fisher information matrix  2  ∂ ` (α, θ | x) , s, k = 1, 2. Is,k = −E ∂α∂θ

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It is clear that there is a difficulty to get the exact expressions of the above expectation. Therefore, we will take the approximate asymptotic variance–covariance matrix for MLE. The approximate asymptotic variance–covariance matrix is given by

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I0 (ˆ α, ˆθ) =

"

2

− ∂ `(α,θ|x) ∂α2 ∂ 2 `(α,θ|x) − ∂θ∂α

− −

∂ 2 `(α,θ|x) ∂α∂θ ∂ 2 `(α,θ|x) ∂θ2

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(ˆ α,ˆ θ)

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As the standard regularity conditions for the asymptotic properties of MLE are satisfied by the exponentiated Weibull distribution (see, Qian [17]). The asymptotic normality of the MLE tell us that (ˆ α, ˆθ) is approximately bivariately normal with mean (α, θ) and covariance matrix   ˆ ˆ ˆ I0 (ˆ α, θ), i.e. (ˆ α, θ) ∼ N (α, θ) , I0 (ˆ α, θ) . Thus, it can be used to compute the 100(1-γ)% approximate confidence intervals for the parameters α and θ, which become, respectively, 

α ˆ − z γ2

p

  q q  p ˆ ˆ ˆ ˆ var(ˆ α), α ˆ + z γ2 var(ˆ α) and θ − z γ2 var(θ), θ + z γ2 var(θ)

(16)

where var(ˆ α) and var(ˆθ) are the elements on the main diagonal of the covariance matrix I0 (ˆ α, ˆθ) γ and z γ2 is the percentile of the standard normal distribution with right-tail probability 2 . 8

ACCEPTED MANUSCRIPT confidence intervals for S(t) and h(t) In order to find the approximate estimates of the variance of Sˆ(t) and Hˆ(t), we use the delta ∂S(t) ∂H(t) method, see Greene [18] and Agresti [19]. Let G1 = ( ∂S(t) ) and G2 = ( ∂H(t) ), where ∂α ∂θ ∂α ∂θ ∂S(t) ∂S(t) ∂h(t) , ∂θ , ∂α ∂α

and ∂h(t) are the first derivatives of the S(t) and h(t) with respect to the param∂θ ˆ ˆ eters α and θ, respectively. The approximate asymptotic variances of S(t) and h(t) are given, respectively, by

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        ˆ ˆ Va ˆr S(t) ' GT1 I0 G1 (ˆα,ˆθ) , and V a ˆr h(t) ' GT2 I0 G2 (ˆα,ˆθ) .

Where, I0 is the approximate asymptotic variance–covariance matrix given in subsection (4.1), and GTl is the transpose of Gl , l = 1, 2. Theses results yields the approximate confidence intervals for S(t) and H(t) as 

and

 q q ˆ ˆ ˆ γ var(S(t)), S(t) + z 2 var(S(t))

  q q ˆ ˆh(t) + z γ var(h(t)) ˆ ˆh(t) − z γ var(h(t)), 2 2

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ˆ − zγ S(t) 2

(17)

(18)

Bootstrap confidence intervals

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It is will known that the normal approximations are adequate when the effective sample size is large enough. However, when the effective sample size m is small, the normal approximations may not work well. Resampling techniques such as the bootstrap may provide more accurate approximate confidence intervals. In this section, we propose to use two confidence intervals based on the parametric bootstrap methods: (i) percentile bootstrap method (PB) based on the idea of Efron [20] and (ii) bootstrap-t method (BT) based on the idea of Hall [21].We illustrate briefly how to estimate confidence intervals of the unknown quantities using both methods. (i) percentile bootstrap (PB) (1) Collect the adaptive progressive Type-II censored data and compute the MLE of Θ = ˆ ˆ = (ˆ ˆ (α, θ, S(t), h(t)), say Θ α, ˆθ, S(t), h(t)), as described in section (2). (2) Based on the pre-specified adaptive progressive censoring scheme (R1 , ..., Rm ) and the switching time T , generate an adaptive Type-II progressive censoring sample from the EW distribution with parameter (ˆ α, ˆθ), using the algorithm described in Ng et al. [13]. (3) Using the bootstrap sample in (2) to obtain the MLE of Θ and denote this bootstrap ∗ ˆ ∗ (t)). ˆ ∗ = (ˆ estimates by Θ α∗ , ˆθ , Sˆ∗ (t), h ∗ ˆ ∗ (t)), i = ˆ ∗1 , Θ ˆ ∗2 , ..., Θ ˆ ∗ where Θ ˆ ∗i = (ˆ (4) Repeat Steps (2)–(3) B times and obtain Θ α∗i , ˆθi , Sˆi∗ (t), h i B 1, 2, ..., B. ˆ ∗i in ascending orders to obtain bootstrap samples: (5) Arrange all component of Θ ∗

∗

∗

∗ ∗ ∗ ˆ ∗ , ..., h ˆ ∗ ). ˆ∗ , h (ˆ α∗(1) , α ˆ ∗(2) , ..., α ˆ ∗(B) ), (ˆθ(1) , ˆθ(2) , ..., ˆθ(B) ), (Sˆ(1) , Sˆ(2) , ..., Sˆ(B) ) and (h (1) (2) (B)

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(ii) bootstrap-t (BT) To obtain the BT confidence intervals for the parameters α and θ, the following algorithm can be applied. Algorithm 1. (1)–(5) the same as the PB above. ∗ ∗ ∗ ˆ ∗(B) ) and (ˆθ(1) , ˆθ(2) , ..., ˆθ(B) ) in step (5).Compute the t-statistics: ˆ ∗(2) , ..., α (6) Using (ˆ α∗(1) , α h ∗ i q  p  ∗ ∗ ∗ ˆ ˆ ˆ ) / V ar(ˆ Tα(j) = (ˆ α(j) − α α)] and Tθ(j) = (θ(j) − θ) / V ar(ˆθ)], j = 1, 2, ..., B.

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(7) For the T ∗ values obtained in step (6), determine the two-sided 100(1 − γ)% BT confidence intervals for α and θ as follows: let Fk (x) = P (Tk∗ ≤ x) be the cumulative distribution functions of T ∗ , where k = ∗ ∗ 1, 2., T1∗ = Tα(j) and T1∗ = Tθ(j) . For a given x, define: q p −1 −1 ˆ α) and θBT (x) = θ + F2 (x) V ar(ˆθ) αBT (x) = α ˆ + F1 (x) V ar(ˆ

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(8) A two-sided 100(1 − γ)% approximate BT confidence intervals for α and θ are given, respectively, by γ γ γ γ (20) (αBT ( ), αBT (1 − )) and (θBT ( ), θBT (1 − )) 2 2 2 2 Algorithm 2. (1)–(5) the same as the PB above. ∗ ∗ ∗ ˆ ∗ ) in step (5).Compute the t-statistics: ˆ ∗ , ..., h ˆ∗ , h ) and (h , ..., Sˆ(B) , Sˆ(2) (6) Using (Sˆ(1) (B) (2) (1) h√ i i h√ ∗ ˆ ˆ ∗ − h(t)) ˆ − S(t)) m(Sˆ(j) m(h (j) ∗ ∗ q q TS(j) = and Th(j) , j = 1, 2, ..., B. = ˆ ˆ V ar(S(t)) V ar(h(t))

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(7) For the T ∗ values obtained in step (6), determine the two-sided 100(1 − γ)% BT confidence intervals for S(t) and h(t) as follows: let Gk (x) = P (Tk∗ ≤ x) be the cumulative distribution functions of T ∗ , ∗ ∗ where k = 1, 2., T1∗ = TS(j) and T1∗ = Th(j) . For a given x, define: s s ˆ ˆ V ar(S(t)) ˆ + G−1 (x) V ar(h(t)) ˆ + G−1 (x) SBT (x) = S(t) and h (x) = h(t) BT 1 2 m m ˆ ∗ (t) as estimates of V ar(S(t)) ˆ We use the sample variance of the bootstrap samples of Sˆi∗ (t) and h i ˆ and V ar(h(t)), respectively. (8) A two-sided 100(1 − γ)% approximate BT confidence intervals for S(t) and h(t) are given, respectively, by γ γ γ γ (21) (SBT ( ), SBT (1 − )) and (hBT ( ), hBT (1 − )) 2 2 2 2 10

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6

Bayes estimations

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Besides being an alternative analysis, the use of the Bayesian method allows the incorporation of previous knowledge of the parameters through informative priori densities. When this information is not available, noninformative priori are considered. In the Bayesian approach, the information referring to the model parameters is obtained through posterior marginal distributions. This section deals with finding Bayes estimates for unknown parameters α, θ, the reliability function S(t) and the hazard function h(t). For estimating these quantities, we assume mainly a squared error loss (SEL) function only; however, any other loss function can be easily incorporated. Since α and θ are both unknown, a natural choice for the prior distributions of α and θ would be to assume that the two quantities are independent gamma G(a, b) and G(c, d) distributions, respectively. As a consequence, the joint prior distribution is given by π(α, θ) ∝ αa−1 e−b α θc−1 e−d θ ,

α > 0, θ > 0, a > 0, b > 0, c > 0, d > 0.

π(α, θ|x) ∝ α

m+a−1 m+c−1

θ

AN US

Subsequently, the joint posterior distribution of α and θ becomes: −b α −d θ

e

e

(1 −

θ Cj Um )

m Y

Vi Uiθ−1

i=1

The posterior density (22) can be rewritten as

i ζR i .

(22)

i=1

(23)

M

π(α, θ|x) ∝ g1 (α|θ, x)g2 (θ|α, x)D(α, θ|x),

j Y

ED

Where the conditional posterior distributions g1 (α|θ,data) and g2 (θ|α,data) of the parameters α and θ can be computed and written, respectively, as

PT

g1 (α|θ, x) ∝ α

m+a−1

−b α

e

CE

D(α, θ|x) = (1 −

θ Cj Um )

Vi

(24)

i=1

g2 (θ|α, x) ∝ θm+c−1 e−d

and

m Y

m Y i=1

θ

Uiθ−1

j Y

(25)

i ζR i

(26)

i=1

AC

Therefore, the Bayes estimate of any function of α and θ, say u(α, θ) under the squared error loss function is R∞R∞ u(α, θ)g1 (α|θ, x)g2 (θ|α, x)D(α, θ|x)dαdθ uˆB (α, θ) = 0 R 0∞ R ∞ (27) g1 (α|θ, x)g2 (θ|α, x)D(α, θ|x)dαdθ 0 0

It is not possible to compute (27) analytically, we propose to approximate it by using an MCMC with importance sampling technique. We generate MCMC samples and then use it to compute desired Bayes estimates and also construct the corresponding confidence intervals. The details are explained below. 11

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We need the following theorem for further development. Theorem 1. The density function g1 (α|θ,data) as given in (24) has a log-concave density function. Proof: Its easy to proof that: m

d2 (m + a − 1) X α 2 log[g1 (α|θ, x)] = − − xi log xi < 0. 2 dα α2 i=1

AN US

CR IP T

Since g1 (α|θ, x) has a log-concave density, using the idea of Devroye [22], it is possible to generate a sample from g1 (α|θ, x). Moreover, since g2 (θ|α, x) follows gamma(m + c, d), it is quite simple to generate from g2 (θ|α, x). Using Theorem 1, Samples are generated using the following Algorithm: Step 1: Generate α from g1 (α|θ, x) using the method developed by Devroye [22]. Step 2: Generate θ from g2 (θ|α, x). Step 3: Repeat Step 1 and Step 2 to generate {(αi , θi ), i = 1, 2, ..., M }. Next, we observe that the approximate Bayes estimate of any function u(α, θ) ≡ Θ ≡ (α, θ, S(t), h(t)) with respect to the squared error loss function is given by

ˆ = uˆB (α, θ) = Θ

1 M

M P

i=1

u(αi , θi )D(αi , θi | x)

1 M

M P

i=1

(28)

D(αi , θi | x)

ED

M

The 100(1 − γ)% credible intervals for α, θ, S(t) and h(t) are constructed using the method of Chen and Shao [23]. We briefly discuss the method below. Let π(Θ|x) and Π(Θ|x) denote the posterior density and posterior distribution functions of Θ , respectively, where Θ is some unknown parameter of interest. Also, let Θ(γ) be the γ − th quantile of Θ, i.e. Θ(γ) = inf{Θ; Π(Θ| x) ≥ γ},

0 < γ < 1.

CE

PT

For a given Θ∗ , a simulation consistent estimator of Π(Θ∗ | x) is given by ∗

Π(Θ |x) =

M P

i=1

IΘ≤Θ∗ D(αi , θi | x) M P

i=1

AC

where IΘ≤Θ∗ is the indicator function defined as ( 1, IΘ≤Θ∗ = 0, The corresponding estimate is obtained as   0,    P i Π(Θ∗ |x) = Ωk ,  k=1    1,

(29)

D(αi , θi | x)

If Θ ≤ Θ∗ If Θ > Θ∗

if Θ∗ < Θ(1) if Θ(i) ≤ Θ∗ < Θ(i+1) if Θ∗ ≥ Θ(M ) 12

(30)

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where

Ωi =

D(αi ,θi | x) , M P D(αi ,θi | x)

and for i = 1, 2, ...M.Θ(i) represents ordered values of Θi . Finally, an

i=1

estimate of Θ(γ) is given by Θ

=



Θ(1) ,

if

i−1 P

k=1

Then, the 100(1 − γ)% credible intervals for Θ is

Ωk

If γ = 0 i P <γ≤ Ωk

] ) ˆ ( Mk ) , Θ ˆ ( K+[(1−γ)M M (Θ ),

(31)

k=1

CR IP T

(γ)

  Θ(1) ,

(32)

with k = 1, 2, ..., M − [(1 − γ)M ]. Here, [.] denotes the greatest integer function.

7

Illustrative examples

AC

CE

PT

ED

M

AN US

To illustrate the inferential procedures developed in the preceding sections, we use the data set from Nichols and Padgett [24]. A complete sample from the data gives 100 observations on breaking stress of carbon fibres (in Gba) are: {0.39, 0.81, 0.85, 0.98, 1.08, 1.12, 1.17, 1.18, 1.22, 1.25, 1.36, 1.41, 1.47, 1.57, 1.57, 1.59, 1.59, 1.61, 1.61, 1.69, 1.69, 1.71, 1.73, 1.8, 1.84, 1.84, 1.87, 1.89, 1.92, 2, 2.03, 2.03, 2.05, 2.12, 2.17, 2.17, 2.17, 2.35, 2.38, 2.41, 2.43, 2.48, 2.48, 2.5, 2.53, 2.55, 2.55, 2.56, 2.59, 2.67, 2.73, 2.74, 2.76, 2.77, 2.79, 2.81, 2.81, 2.82, 2.83, 2.85, 2.87, 2.88, 2.93, 2.95, 2.96, 2.97, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19, 3.19, 3.22, 3.22, 3.27, 3.28, 3.31, 3.31, 3.33, 3.39, 3.39, 3.51, 3.56, 3.6, 3.65, 3.68, 3.68, 3.68, 3.7, 3.75, 4.2, 4.38, 4.42, 4.7, 4.9, 4.91, 5.08, 5.56}. Qian [17] used the standard likelihood ratio test to show that the EW distribution is acceptable for modeling these breaking stress data. Example: In this example we consider the case when the data are adaptive Type-II progressive censored. It is assumed that we observe only 60 data points and the rest are progressive censored. In this case we take two distinct values of T , T = (1.40, 3.33) and R = ( 20, 058 , 20).For brevity, for example, the censoring scheme R = (2, 30 , 5) is denoted by R = (2, 0, 0, 0, 5).Thus, the adaptive Type-II progressive censored samples are: For (T = 1.40) : 0.39, 0.85, 0.98, 1.12, 1.17, 1.18, 1.22, 1.36, 1.41, 1.57, 1.57, 1.59, 1.61, 1.61, 1.69, 1.69, 1.71, 1.73, 1.80, 1.84, 1.84, 1.87, 1.92, 2.03, 2.03, 2.12, 2.17, 2.17, 2.17, 2.35, 2.38, 2.41, 2.48, 2.48, 2.5, 2.53, 2.55, 2.55, 2.56, 2.59, 2.67, 2.74, 2.77, 2.79, 2.81, 2.82, 2.83, 2.87, 2.88, 2.93, 2.95, 2.96, 2.97, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19. In this censoring scheme we observe that (j = 8). For (T = 3.33) : 0.39, 0.81, 0.98, 1.08, 1.12, 1.18, 1.22, 1.25, 1.36, 1.47, 1.57, 1.57, 1.59, 1.61, 1.61, 1.69, 1.69, 1.71, 1.73, 1.84, 1.84, 1.87, 2.0, 2.03, 2.03, 2.05, 2.12, 2.17, 2.17, 2.35, 2.38, 2.43, 2.48, 2.48, 2.5, 2.53, 2.55, 2.55, 2.56, 2.59, , 2.73, 2.74, 2.76, , 2.81, 2.81, 2.82, 2.83, 2.87, 2.88, 2.93, 2.95, 2.97, 2.97, 3.09, 3.11, 3.15, 3.19, 3.19, 3.22, 3.22. In this censoring scheme we observe that (j = 60, i.e. Xm:m:n < T ). 13

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CR IP T

The MLEs of the unknown quantities for complete sample (n = m = 100) are: (ˆ αM L , ˆθM L ) = ˆ = 1.6)) = (0.826088 , 0.42742).Based on the above ˆ = 1.6), h(t (1.02514, 7.92322) and (S(t adaptive Type-II progressive censored data we compute different estimations (ML, PB and BT) and the corresponding 95% confidence intervals of θ, α, S(t) and h(t). For Bayes estimation, it is assumed that both the parameters are unknown. Since we do not have any prior information available, we use non-informative priors on both α and θ, (a = b = c = d = 0).Using Algorithm in the previous Section, we generate 1000 MCMC samples {(αi , θi ), i = 1, 2, ..., 1000} and based on them we compute the approximate Bayes MCMC estimates. The results are listed in Tables 1 and 2. Table 1: Point estimations of α, θ, S(t) and h(t) with t = 1.6

T = 1.41 ML

PB

BT

MCMC

ML

1.0624 7.7796 0.8105 0.4744

1.0712 8.0067 0.8137 0.4730

1.0705 7.8029 0.81310 0.4730

1.0578 7.7379 0.8067 0.4757

1.0157 7.9485 0.8295 0.4167

PB

BT

MCMC

1.0213 8.0829 0.8295 0.4178

1.0194 7.9490 0.8321 0.4134

1.0100 7.8561 0.8239 0.4202

AN US

α θ S(t) H(t)

T = 3.33

Table 2: 95% confidence intervals of α, θ, S(t) and h(t) with t = 1.6

T = 1.41 Interval

CE AC

BT

M CM C

M

(0.9580, 1.1668) (6.0067, 9.5524) (0.7465, 0.8745) (0.3578, 0.591) (0.9698, 1.1860) (6.3879, 10.0671) (0.7495, 0.8726) (0.3652, 0.6091) (0.9653, 1.1699) (5.7714, 9.4635) (0.7477, 0.8883) (0.3391, 0.5878) (0.9503, 1.1608) (6.0722, 9.6071) (0.7382, 0.8659) (0.3673, 0.5934)

PT

PB

α θ S(t) h(t) α θ S(t) h(t) α θ S(t) h(t) α θ S(t) h(t)

ED

ML

T = 3.33.

Length

Interval

Length

0.2088 3.5457 0.1280 0.2332 0.2162 3.6792 0.1232 0.2439 0.2046 3.6922 0.1406 0.2487 0.2104 3.5349 0.1278 0.2261

(0.9181, 1.1132) (6.1494, 9.6476) (0.7678, 0.8911) (0.3094, 0.524) (0.9252, 1.1314) (6.4957, 10.1179) (0.7720, 0.8910) (0.3102, 0.5270) (0.9139, 1.1202) (6.0308, 9.6582) (0.7686, 0.9002) (0.2991, 0.5232) (0.9118, 1.1060) (6.1642, 9.7256) (0.7549, 0.8825) (0.3200, 0.5230)

0.1951 3.4982 0.1233 0.2146 0.2062 3.6222 0.1190 0.2168 0.2063 3.6274 0.1316 0.2241 0.1943 3.5614 0.1277 0.2030

14

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8

Monte Carlo Simulation and Comparisons

Conclusion

AC

9

CE

PT

ED

M

AN US

CR IP T

In this section, a Monte Carlo simulation study is carried out in order to compare the proposed estimates of α, θ, S(t) and h(t). Comparison between different estimators is made with respect to their MSE values. In computing the estimates we generated 1000 samples (adaptive type-II progressive censored samples) from the EW distribution by using the algorithm described in Ng et al. [13], and we replicated the process 1000 times. The associated MLEs are computed using a numerical technique. Bayes MCMC estimates of unknown quantities are derived with respect to the squared error loss function. The true value of (α, θ) is taken as (1.2, 2) and T = (0.9, 2.0).For prior information we have used: Non-informative prior, Prior 1 with a = b = c = d = 0, and informative prior, Prior 2 with a = 1.2, b = 1, c = 2, d = 1. For Prior 2 we have chosen the hyperparameters in such a way that the prior mean became the expected value of the corresponding parameter. We perform a Monte Carlo Simulation study using different sample sizes (n), different effective sample sizes (m). The following censoring schemes (CS) are considered. scheme I: R1 = n − m, Ri = 0 for i 6= 1. scheme II: R m2 = n − m, Ri = 0 for i 6= m2 ; if m is even, and R m+1 = n − m, Ri = 0 for i 6= m+1 ; if m is odd. 2 2 scheme III: Rm = n − m, Ri = 0 for i 6= m. The averages and MSEs in parentheses of the estimators of α, θ, S(t) and h(t) for different sample sizes, censoring schemes and T = 0.9 and 2.0 are presented in Tables 3. In addition, the average estimates and MSEs of S(t) and h(t) are computed for t = 0.75,the true values are S(t) = 0.742541 and h(t) = 0.762689, The results are tabulated in Table 4. For each simulated sample we computed 95% confidence intervals and checked whether the true value lay within the interval and recorded the length of the confidence interval. This procedure was repeated 1000 times. The estimated coverage probability was computed as the number of confidence intervals that covered the true values divided by 1000, whereas the estimated expected width of the confidence interval was computed as the sum of the lengths for all intervals divided by 1000. The expected widths and the coverage probabilities in parentheses are presented in Tables 5 and 6.

In this paper, we have considered the Bayes and non-Bayes estimations of the parameters, reliabilty and hazard functions of the exponentiated Weibull distribution based an adaptive Type-II progressive censoring scheme. This censoring scheme allows us to choose the next censoring number taking into account both the previous censoring numbers and previous failure times. The MLEs, the bootstrap confidence intervals and the asymptotic confidence intervals based on the observed Fisher information matrix have been discussed. We assume the gamma priors for both the unknown parameters and provide the Bayes estimators under the assumptions of squared error 15

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Table 3. Average estimates and the MSEs of α and θ for different methods n m CS MLE (MCMC-Prior 0) (MCMC-Prior 1)

50

40

I II III

60

40

I II III

50

30

I II III

50

40

I II

60

40

1.9865 (0.0997) 2.0103 (0.0945) 1.9817 (0.0949) 2.0014 (0.0866) 2.0074 (0.0797) 2.0112 (0.0836) 2.0003 (0.0916) 2.0151 (0.0737) 1.9917 (0.0716)

CE

III

1.2310 (0.0247) 1.2503 (0.0326) 1.2832 (0.0622) 1.2310 (0.0194) 1.2374 (0.0206) 1.2432 (0.0296) 1.2401 (0.0203) 1.2397 (0.0264) 1.2554 (0.0373)

I

AC

II

III

θ

1.9657 (0.1089) 1.9794 (0.0815) 1.9688 (0.0769) 1.9708 (0.0584) 1.9703 (0.0609) 1.9707 (0.0628) 1.984 (0.06270) 1.9956 (0.0598) 1.9900 (0.0608)

1.2292 (0.0238) 1.2307 (0.0321) 1.2671 (0.0593) 1.2232 (0.0169) 1.2295 (0.025) 1.2379 (0.0271) 1.2214 (0.0173) 1.2336 (0.0263) 1.2403 (0.0326)

1.9624 (0.0626) 1.9813 (0.0651) 1.9733 (0.0683) 1.9734 (0.0547) 1.9721 (0.0566) 1.9731 (0.0584) 1.9866 (0.0585) 1.9969 (0.0560) 1.9920 (0.0564)

1.2162 (0.0226) 1.2328 (0.0287) 1.2715 (0.0555) 1.2184 (0.0179) 1.2240 (0.0187) 1.2308 (0.027) 1.2280 (0.0186) 1.2269 (0.0241) 1.2460 (0.0340)

1.9598 (0.0643) 1.9911 (0.0637) 1.9662 (0.0701) 1.9727 (0.0581) 1.9852 (0.0552) 1.9848 (0.0578) 1.9750 (0.0611) 2.0010 (0.0539) 1.9799 (0.0550)

CR IP T

III

2.0032 (0.1003) 1.9949 (0.0914) 1.9923 (0.0962) 2.0038 (0.0849) 1.9931 (0.0784) 1.9976 (0.083) 2.0146 (0.0885) 2.0099 (0.0757) 2.0082 (0.0767)

α

AN US

II

1.2445 (0.0263) 1.2483 (0.0366) 1.2793 (0.0675) 1.2359 (0.0184) 1.2423 (0.0274) 1.2504 (0.0297) 1.2331 (0.0188) 1.2462 (0.0288) 1.2501 (0.0354)

θ

M

I

α T = 0.9 1.2309 (0.0251) 1.2333 (0.0352) 1.2762 (0.0715) 1.2239 (0.0175) 1.2308 (0.0264) 1.2398 (0.0288) 1.2222 (0.0180) 1.2352 (0.0278) 1.2435 (0.0356) T =2 1.2178 (0.024) 1.2352 (0.0311) 1.2806 (0.0653) 1.2192 (0.0186) 1.2252 (0.0197) 1.2327 (0.0289) 1.2291 (0.0194) 1.2282 (0.0254) 1.2494 (0.0376)

ED

30

θ

PT

50

α

1.9562 (0.0704) 1.9891 (0.0695) 1.9616 (0.0787) 1.9697 (0.0622) 1.9833 (0.0592) 1.9824 (0.0622) 1.9723 (0.0658) 2.0003 (0.0574) 1.9774 (0.0596)

16

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Table 4. Average estimates and the MSEs of S(t) and h(t) for different methods S(t) 30

I II III

50

40

I II III

60

40

I II III

50

30

I II III

50

40

I

0.7373 (0.0027) 0.7437 (0.0025) 0.7418 (0.0023) 0.7407 (0.0025) 0.7429 (0.0021) 0.7438 (0.0023) 0.7412 (0.0025) 0.7448 (0.0019) 0.7422 (0.0018)

CE

II

0.7415 (0.0028) 0.7409 (0.0024) 0.7430 (0.0023) 0.7418 (0.0024) 0.7408 (0.0022) 0.7421 (0.0023) 0.7431 (0.0025) 0.7444 (0.0019) 0.7443 (0.0019)

PT

50

AC

III

60

40

I

II

III

(MCMC-Prior 0) h(t) S(t) h(t) T = 0.9 0.7799 0.7294 0.7905 (0.0169) (0.0022) (0.0145) 0.7836 0.7326 0.7866 (0.0193) (0.002) (0.0174) 0.7921 0.7342 0.8003 (0.0257) (0.0019) (0.0256) 0.7756 0.7312 0.7846 (0.0121) (0.0019) (0.0102) 0.7809 0.7320 0.7863 (0.0146) (0.0018) (0.0132) 0.7816 0.7329 0.7886 (0.015) (0.0019) (0.0138) 0.7717 0.7333 0.7803 (0.0129) (0.0020) (0.0101) 0.7767 0.7375 0.7797 (0.0141) (0.0016) (0.0129) 0.7779 0.7371 0.7830 (0.0156) (0.0016) (0.0149) T =2 0.7817 0.7262 0.7899 (0.0172) (0.0022) (0.0148) 0.7788 0.7347 0.7836 (0.0175) (0.002) (0.0154) 0.7976 0.7334 0.8049 (0.025) (0.0019) (0.0249) 0.7751 0.7303 0.7837 (0.0129) (0.0021) (0.0111) 0.7754 0.7339 0.7813 (0.013) (0.0017) (0.0114) 0.775 0.7343 0.7824 (0.0145) (0.0018) (0.0132) 0.7794 0.7317 0.7871 (0.0145) (0.002) (0.0125) 0.7728 0.7377 0.7761 (0.0132) (0.0015) (0.0119) 0.7847 0.7355 0.7889 (0.0160) (0.0016) (0.0154)

(MCMC-Prior 1) S(t) h(t) 0.7305 (0.0020) 0.7333 (0.0018) 0.7350 (0.0017) 0.7320 (0.0018) 0.7326 (0.0017) 0.7335 (0.0017) 0.7341 (0.0019) 0.7379 (0.0015) 0.7376 (0.0015)

0.7878 (0.0130) 0.7843 (0.0155) 0.7951 (0.0211) 0.7830 (0.0095) 0.7847 (0.0122) 0.7866 (0.0125) 0.7786 (0.0101) 0.7783 (0.0119) 0.7808 (0.0134)

CR IP T

MLE

AN US

C.S

M

m

ED

n

17

0.7273 (0.002) 0.7354 (0.0018) 0.7341 (0.0017) 0.7311 (0.0019) 0.7345 (0.0016) 0.7349 (0.0017) 0.7325 (0.0019) 0.7381 (0.0015) 0.7360 (0.0015)

0.7872 (0.0133) 0.7813 (0.0138) 0.7995 (0.021) 0.7820 (0.0103) 0.7798 (0.0105) 0.7805 (0.012) 0.7852 (0.0115) 0.7749 (0.0110) 0.7864 (0.0137)

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Table 5. The expected width of 95% confidence intervals and coverage probabilities for α and θ n m C.S MLE (MCMC-Prior 0) (MCMC-Prior 1)

II III 50

40

I II III

60

40

I II III

50

30

I II III

40

I

0.5784 (0.9520) 0.6453 (0.9500) 0.8713 (0.9610) 0.4999 (0.9440) 0.5340 (0.9630) 0.6009 (0.950) 0.5037 (0.5037) 0.5536 (0.9480) 0.6847 (0.9610)

PT

50

0.5832 (0.9540) 0.6539 (0.9500) 0.8667 (0.9580) 0.5019 (0.9590) 0.5813 (0.9430) 0.6063 (0.9390) 0.4988 (0.9420) 0.5926 (0.9520) 0.6787 (0.9640)

II

CE

III

60

40

I

AC

II

III

α

θ

1.1998 (0.9760) 1.1360 (0.9650) 1.1355 (0.9570) 1.1022 (0.9790) 1.0567 (0.9630) 1.0532 (0.9690) 1.0857 (0.9700) 1.0217 (0.9640) 1.0177 (0.9640)

0.5708 (0.9490) 0.6348 (0.9500) 0.8377 (0.9490) 0.4939 (0.9610) 0.5693 (0.9410) 0.5936 (0.9360) 0.4905 (0.9460) 0.5791 (0.9460) 0.6631 (0.9560)

1.1767 (0.9750) 1.1138 (0.9690) 1.1118 (0.9660) 1.0833 (0.9830) 1.0401 (0.9710) 1.0355 (0.9730) 1.0681 (0.9730) 1.0057 (0.9660) 1.0014 (0.9700)

1.2075 (0.9750) 1.1315 (0.9690) 1.1369 (0.9540) 1.102 (0.9630) 1.0631 (0.9660) 1.0563 (0.9640) 1.0857 (0.9710) 1.0236 (0.9680) 1.018 (0.9660)

0.5646 (0.9480) 0.6278 (0.9540) 0.8411 (0.9520) 0.4918 (0.9490) 0.5245 (0.9600) 0.5887 (0.9480) 0.4950 (0.9520) 0.5422 (0.9420) 0.6686 (0.9540)

1.1803 (0.9790) 1.1092 (0.9700) 1.1096 (0.9620) 1.0844 (0.9740) 1.0473 (0.9630) 1.0408 (0.9710) 1.0682 (0.9740) 1.0062 (0.9710) 1.0008 (0.9700)

CR IP T

I

θ

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α T = 0.9 1.3736 0.5770 (0.9570) (0.9460) 1.2250 0.6448 (0.9460) (0.9390) 1.2186 0.8657 (0.9430) (0.9430) 1.2333 0.4974 (0.9700) (0.9590) 1.1462 0.5762 (0.9430) (0.9390) 1.1481 0.6005 (0.9530) (0.9350) 1.2171 0.4949 (0.9540) (0.9440) 1.0878 0.5867 (0.9460) (0.950) 1.085 0.6739 (0.9530) (0.9520) T =2 1.3628 0.5721 (0.9610) (0.9470) 1.2336 0.6366 (0.9520) (0.9450) 1.2146 0.8701 (0.9450) (0.9450) 1.2314 0.4955 (0.953) (0.9450) 1.1601 0.5287 (0.951) (0.9540) 1.1545 0.5954 (0.951) (0.9430) 1.2107 0.4998 (0.9560) (0.9470) 1.0965 0.5474 (0.9520) (0.9430) 1.0784 0.6798 (0.9540) (0.9510)

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Table 6. The expected width of 95% confidence intervals and coverage probabilities for S(t) and h(t) n m C.S MLE (MCMC-Prior 0) (MCMC-Prior 1)

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I II III

60

40

I II III

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I II III

40

I

0.2297 (0.9760) 0.1979 (0.9390) 0.1935 (0.9590) 0.2080 (0.9650) 0.1932 (0.9520) 0.1919 (0.9460) 0.2034 (0.9520) 0.1781 (0.9460) 0.1762 (0.9620)

PT

50

0.2287 (0.9770) 0.1986 (0.9490) 0.1932 (0.9470) 0.2080 (0.9620) 0.1930 (0.9640) 0.1922 (0.9560) 0.2028 (0.9550) 0.1776 (0.9520) 0.1757 (0.9450)

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III

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III

S(t)

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h(t)

0.4831 (0.9730) 0.4864 (0.9620) 0.5467 (0.9510) 0.4273 (0.9710) 0.4337 (0.9630) 0.4402 (0.9650) 0.4225 (0.9670) 0.4282 (0.9370) 0.4522 (0.9600)

0.2034 (0.9790) 0.1839 (0.9730) 0.1796 (0.975) 0.1894 (0.981) 0.1794 (0.9750) 0.1782 (0.9760) 0.1840 (0.9710) 0.1667 (0.9730) 0.1649 (0.979)

0.4718 (0.9760) 0.4751 (0.9650) 0.5292 (0.960) 0.4193 (0.9760) 0.4253 (0.9690) 0.4316 (0.9670) 0.4147 (0.9720) 0.4204 (0.9460) 0.4429 (0.9640)

0.4845 (0.9710) 0.4821 (0.9620) 0.5517 (0.9520) 0.4268 (0.9690) 0.4242 (0.9720) 0.4367 (0.9560) 0.4267 (0.9600) 0.4205 (0.9480) 0.4564 (0.9520)

0.2057 (0.9800) 0.1822 (0.9760) 0.1800 (0.9720) 0.1898 (0.9740) 0.1794 (0.9740) 0.1780 (0.9770) 0.1853 (0.9770) 0.1666 (0.9810) 0.1661 (0.9820)

0.4718 (0.9740) 0.4709 (0.9650) 0.5322 (0.9550) 0.4184 (0.9770) 0.4168 (0.9700) 0.9770 (0.9600) 0.4186 (0.9620) 0.4122 (0.9590) 0.4466 (0.9560)

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h(t)

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S(t) T = 0.9 0.5207 0.2076 (0.9720) (0.9770) 0.5082 0.1871 (0.9500) (0.9710) 0.5626 0.1825 (0.9510) (0.9730) 0.4561 0.1931 (0.9600) (0.9780) 0.452 0.1823 (0.9630) (0.9700) 0.4593 0.1811 (0.9550) (0.9740) 0.4510 0.1872 (0.9620) (0.9700) 0.4427 0.1691 (0.9380) (0.9760) 0.4652 0.1673 (0.9530) (0.9760) T =2 0.5184 0.2102 (0.9670) (0.9780) 0.5059 0.1855 (0.9550) (0.9730) 0.5665 0.1833 (0.9520) (0.9710) 0.4549 0.1932 (0.9700) (0.9700) 0.4454 0.1820 (0.9640) (0.9720) 0.4560 0.1806 (0.9480) (0.9760) 0.4538 0.1884 (0.9550) (0.9750) 0.4375 0.1693 (0.9450) (0.9800) 0.4689 0.1683 (0.9530) (0.9770)

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loss functions. We found that when both parameters are unknown, the Bayes estimates cannot be obtained in explicit form. We used the Gibbs sampling technique to generate MCMC samples and then using importance sampling methodology to compute the approximate Bayes estimates and constructed the credible intervals. The details have been explained using a real life example. The performance of different methods was compared via a Monte Carlo simulation. From Tables, it can be seen that the performance of the MLEs is quite close to that of the Bayes estimators based on the noninformative priors, as expected. Thus, if we have no prior information on the unknown parameters, then it is always better to use the MLEs rather than the Bayes estimators, because the Bayes estimators are computationally more expensive. Also, in comaring the schemes I and III, it is clear that the MSEs of the MLEs and Bayes estimators are greater for the censoring scheme III than the censoring scheme I. This may not be surprising, because the expected duration of the experiments for censoring scheme I is greater than the censoring scheme III. Thus the data obtained by the censoring scheme I would be expected to provide more information about the unknown parameters than the data obtained by censoring scheme III. Funding This research was supported by the Institute of Scientific Research and Revival of Islamic Heritage, Umm Al-Qura University, under the project (No. 43305016). Acknowledgment The author would like to express thanks to the Editors and the anonymous referees for their valuable comments and suggestions, which significantly improved the paper.

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