Estimation and prediction for a progressively censored generalized inverted exponential distribution

Statistical Methodology 32 (2016) 185–202

Contents lists available at ScienceDirect

Statistical Methodology journal homepage: www.elsevier.com/locate/stamet

Estimation and prediction for a progressively censored generalized inverted exponential distribution Sanku Dey a , Sukhdev Singh b , Yogesh Mani Tripathi b,∗ , A. Asgharzadeh c a

Department of Statistics, St. Anthony’s College, Shillong-793001, Meghalaya, India

b

Department of Mathematics, Indian Institute of Technology Patna, Bihta-801103, India

c

Department of Statistics, University of Mazandaran, Babolsar, Iran

highlights • Bayesian and classical estimates are derived for the unknown parameters of a generalized inverted exponential distribution based on progressively Type-II censored data.

• Prediction of future failures are discussed using classical and Bayesian approaches. • A simulation study is conducted to assess the behavior of proposed methods and two real data sets are analyzed for illustrative purposes.

article

info

Article history: Received 27 October 2015 Received in revised form 28 May 2016 Accepted 28 May 2016 Available online 21 June 2016 Keywords: Asymptotic confidence interval Bayesian estimation Equal-tail interval HPD interval Maximum likelihood estimation MH algorithm Prediction

∗

abstract In this paper, we consider generalized inverted exponential distribution which is capable of modeling various shapes of failure rates and aging criteria. The purpose of this paper is two fold. Based on progressive type-II censored data, first we consider the problem of estimation of parameters under classical and Bayesian approaches. In this regard, we obtain maximum likelihood estimates, and Bayes estimates under squared error loss function. We also compute 95% asymptotic confidence interval and highest posterior density interval estimates under the respective approaches. Second, we consider the problem of prediction of future observations using maximum likelihood predictor, best unbiased predictor, conditional median predictor and Bayes predictor. The associated predictive interval estimates for the censored observations are computed as well. Finally, we analyze two real data sets and conduct a

Corresponding author. E-mail address: [email protected] (Y.M. Tripathi).

http://dx.doi.org/10.1016/j.stamet.2016.05.007 1572-3127/© 2016 Elsevier B.V. All rights reserved.

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S. Dey et al. / Statistical Methodology 32 (2016) 185–202

Monte Carlo simulation study to compare the performance of the various proposed estimators and predictors. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The one-parameter exponential distribution is the simplest and most widely used lifetime model in life testing and reliability analysis. In spite of its popularity, this distribution has certain restrictions like constant hazard rate. To make its applicability more flexible, a large amount of work has been done via exponentiating the distribution function (see Gupta and Kundu [16]). Another modification to this distribution has been done by using its inverted version, known as the inverted exponential (IE) distribution. Lin et al. [24] studied one-parameter IE distribution and obtained maximum likelihood estimator, confidence limits, and UMVUE for the parameter and the reliability function based on complete samples. They also compared this model with inverted Gaussian and log-normal distributions based on maintenance data set, and observed that it provides a better fit than these two distributions. Further, Dey [10] obtained Bayes estimators of the parameter and risk functions under different loss functions. Abouammoh and Alshingiti [1] introduced a shape parameter to the IE distribution to obtain generalized inverted exponential (GIE) distribution. The probability density function (PDF) and cumulative distribution function (CDF) of GIE(γ , λ) distribution are respectively given by f (x; γ , λ) =

γλ x2

λ

λ γ −1

e− x 1 − e− x



λ

F (x; γ , λ) = 1 − (1 − e− x )γ .



,

x > 0, γ > 0, λ > 0,

(1) (2)

Notice that here γ is the shape parameter and λ is the scale parameter of GIE distribution, and moreover correspond to γ = 1 GIE distribution reduces to IE distribution. It is observed that the hazard function of GIE distribution can be increasing or decreasing but not constant, depending on the shape parameter. It is also observed that in many situations this distribution may provide a better fit than gamma, Weibull, and generalized exponential distributions (see Abouammoh and Alshingiti [1]). For recent contributions to this distribution, one may refer to Krishna and Kumar [19], Dey and Dey [12], Dey and Pradhan [14], Dey and Dey [11], Dey et al. [13], Singh et al. [30], Singh et al. [33], and Dube et al. [15]. In recent past, progressive type-II censoring has received much attention in the literature of life testing and reliability analysis. The main advantage of this censoring over the traditional type-II censoring is that the removal of live units at intermediate stages is allowed under this censoring, whereas, under type-II censoring live units can only be removed at the time of the termination of experiment. Lifetime data under this censoring can be collected in the following way. Suppose that a sample of n independent and identical units is put on a life test experiment. Further assume that life times of the units follow PDF f (x; θ ) and CDF F (x; θ ), here θ is a vector of unknown parameters of the distribution. Now as the experiment will start the units on the test will start failing, let us suppose that first failure occurs at a random time X(1) . Then under this censoring at the time X(1) , R1 number of live units are removed from the remaining n − 1 units in the experiment. In a similar way, when a second failure occurs at a random time X(2) , R2 number of live units are randomly removed from the remaining n − 2 − R1 units, and so on, and at the time of mth failure X(m) , all the

m−1

remaining n − m − i=1 Ri number of units are removed, and the experiment is terminated. Here, the censoring m scheme R = (R1 , R2 , . . . , Rm ) is prefixed prior to the commencement of the experiment so that i=1 Ri = n − m. It is to be noted that corresponding to R1 = R2 = · · · = Rm−1 = 0 and Rm = n − m, this censoring reduces to the traditional type-II censoring. Further, corresponding to m = n and Ri = 0, i = 1, 2, . . . , n, it reduces to complete sample having no censoring. Therefore,

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the type-II and complete samples can also be viewed as a particular case of this censoring. One may refer to the recent book entitled The Art of Progressive Censoring by Balakrishnan and Cramer [7] for extensive literature and applications of this censoring. Estimation of parameters of a distribution and prediction of future observations are main problems in statistical inference. Estimating the parameters of a distribution is an essential topic because many of its features depend on the parameters. On the other hand, the prediction of life-times of censored units is of great significance in statistics in case of censored data. For instance, one could obtain useful information on the total time duration of the life testing experiments by predicting the largest order statistic in a censored sample. This in turn would help the life-tester to plan the future life-test of the unit. Likewise, in case of prediction from progressively censored samples, the life tester may gather useful information regarding additional cost that would be incurred during the life test (e.g., cost of testing or destruction of additional units) if the life-tester were to alter the progressive censoring scheme. Further, this would help in determining the viability of an intended progressively censored life-testing experiment in the future based on time and cost (see, Basak and Balakrishnan [8]). Besides, in clinical and reliability studies, various types of censored data have been observed, and the problems of prediction of censored observations under classical and Bayesian viewpoints have been studied by several authors. See Ali-Mousa [2], Ali-Mousa and Jaheen [4], Jaheen [18], Soliman [35], Ali-Mousa and Al-Sagheer [3], Pradhan and Kundu [27], Kundu and Pradhan [22], Kundu [20], Raqab et al. [28], Huang and Wu [17], and Singh et al. [34], Asgharzadeh et al. [5], Valiollahi et al. [37] and references cited therein. In literature, Krishna and Kumar [19] discussed GIE distribution under progressive type-II censoring. Authors obtained maximum likelihood estimator and associated asymptotic confidence interval estimates for unknown parameters, and also discussed reliability characteristics and other distribution properties. Recently, Dey and Dey [12] obtained Bayes estimates and associated highest posterior density interval estimates. They have also shown the uniqueness and existence of the maximum likelihood estimators. One may further refer to Singh et al. [30] for random removal case. In this paper authors obtained maximum likelihood estimators of unknown parameters and computed Bayes estimators by assuming gamma prior distributions. Performance of proposed estimates is compared using a simulation study and comments are obtained based on this study. Finally a real data set is analyzed for illustrative purposes. To the best of our knowledge, problems of prediction under classical and Bayesian approaches have not yet been considered for GIE distribution under progressive type-II censoring. The aim of the present paper is two fold. First we consider the problem of estimation of parameters under classical and Bayesian approaches for different sample sizes, different effective sample sizes, different priors, and for different sampling schemes. We also compute 95% asymptotic confidence interval and highest posterior density interval estimates under the respective approaches. Second, we consider the problem of prediction and predict the censored observations using maximum likelihood predictor, best unbiased predictor, conditional median predictor, and Bayes predictor based on the assumption that the shape and scale parameters have gamma priors. Further, we also obtain the associated predictive intervals of the censored observations. This would enable us to develop a guideline for choosing the best estimation and prediction methods for the GIE distribution, which we think would be of deep interest to applied statisticians. The rest of this paper is organized as follows. Maximum likelihood estimation is discussed in Section 2. Bayesian estimation is considered in Section 3, and a discussion on elicitation of hyperparameter values is also presented in this section. Section 4 deals with the problem of point prediction in which various point predictors are discussed. In Section 5 various prediction intervals are computed. Finally two real data sets are analyzed and a simulation study is carried out in Section 6. Finally a conclusion is presented in Section 7. 2. Maximum likelihood estimation In this section we obtain maximum likelihood estimators (MLEs) for the unknown parameters of a GIE distribution based on progressive type-II censored data. Suppose that a random sample of n units whose lifetimes follow GIE(γ , λ) distribution is put on a life test experiment. Further, assume

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that x = (x(1) , x(2) , . . . , x(m) ) represent observed lifetimes of the m units under progressive type-II censoring scheme R = (R1 , R2 , . . . , Rm ). Then the associated likelihood function of (γ , λ) given the observed data x can be written as L(γ , λ | x) = C

m 

f (x(i) ; γ , λ)(1 − F (x(i) ; γ , λ))Ri ,

(3)

i=1

where C = n(n − 1 − R1 )(n − 2 − R1 − R2 ) . . . (n − m + 1 − i=1 Ri ); and PDF f (.; γ , λ) and CDF F (·; γ , λ) are respectively given by (1) and (2). By differentiating the associated log-likelihood l = ln L(γ , λ | x) with respect to γ and λ and equating them to zero, the MLE of γ , denoted as γˆ , can  −λ/x(i) ˆ where A(λ) = m . Further the MLE of be obtained as γˆ = −m/A(λ) i=1 (Ri + 1) ln(1 − zi ), zi = e

m−1

λ, denoted by λˆ , is the solution to the equation λ = h(λ), where  −1 m m m  m  1 zi m  Ri zi h(λ) = m + +1 + . x A(λ) x (1 − zi ) A(λ) i=1 x(i) (1 − zi ) i=1 (i) i=1 (i)

Notice that to solve the equation λ = h(λ), first consider λ(0) as an initial guess of λ. Then the successive approximations of the λ can be obtained as λ(1) = h(λ(0) ), λ(2) = h(λ(1) ), . . . , λ(k+1) = h(λ(k) ). This iterative procedure can be terminated at the kth stage if |λ(k+1) − λ(k) | < ϵ , for some small pre-specified value of ϵ . Subsequently, after kth stage the maximum likelihood estimate of λ is ˆ = λ(k) , which can be then plugged into −m/A(λ) ˆ to obtain maximum likelihood estimate given by λ of γ . One may also refer to Dey and Pradhan [14], and Pradhan and Kundu [27] in this regard. For the uniqueness and existence of MLEs we further refer to the work of Dey and Dey [12]. Now the asymptotic variance–covariance matrix of the MLEs of γ and λ can obtained by inverting the observed information matrix, and is given by

σ11 σ21

σ12 σ22



−1 ∂ 2l −E  ∂γ 2∂λ  . ∂ l  −E ∂λ2 γ =γˆ ,λ=λˆ

 ∂ 2l  −E ∂γ 2  2  =  ∂ l −E ∂λ∂γ 







(4)

It is easy to show that

  m  ∂ 2l   e−λ/x(i) = (Ri + 1)E , ∂γ ∂λ x(i) (1 − e−λ/x(i) ) i=1   m  ∂ 2l     m e−λ/x(i) and E =− 2 − γ (Ri + 1) − 1 E 2 . ∂λ2 λ x(i) (1 − e−λ/x(i) )2 i =1 E

 ∂ 2l  m = − 2, 2 ∂γ γ

E

We use the following lemma to compute elements of the Fisher Information matrix. Lemma 1. If X(1) < X(2) < · · · < X(m) is a progressive type-II censored sample from GIE(γ , λ) with censored scheme (R1 , R2 , . . . , Rm ), then e−λ/X(i)

 E

=

X(i) (1 − e−λ/X(i) )

 E



e X(2i)

where ψ(x) =

d dx

λ



−λ/X(i)

(1 −

i ci−1 

=

e−λ/X(i) )2

k=1

ηk = n − k + 1 +

 s=k

Rs

ψ(γ ηk + 1) − ψ(2) ηk (γ ηk − 1)



i  γ ci−1 ai,k [(γ ηk − 1)(γ ηk − 2)[ψ1 (2) 2 λ k=1  − ψ1 (γ ηk ) + (ψ(2) − ψ(γ ηk ))2 ] ,

ln Γ (x) and ψ1 (x) = m

 ai,k

d2 dx2

ln Γ (x) are digamma and trigamma functions and

S. Dey et al. / Statistical Methodology 32 (2016) 185–202

ci−1 =

i 

189

ηs

s=1

ai,k =

k 

1

s=1

ηs − ηi

.

s̸=i

Proof. See Appendix. The two-sided 100(1 −α)%, 0 < α < 1, asymptotic confidence intervals for the parameters γ and √ √ λ can be obtained as γˆ ± z α2 σ11 and λˆ ± z α2 σ22 , respectively, here z α2 is the upper α2 th percentile

of the standard normal distribution. 3. Bayesian estimation

In this section, we discuss the Bayesian inference of the unknown parameters of a GIE distribution. For parameter estimation we have considered squared error loss function. However any other loss function can also be incorporated in the present work. It is observed that if the scale parameter λ is known, the shape parameter γ has a conjugate prior, which is a gamma prior. When both the parameters of the model are unknown, a joint conjugate prior for the parameters does not exist. In view of the above, we propose to use independent gamma priors for both λ and γ (see, Kundu and Howlader [21], Singh et al. [31]) having PDFs

π1 (γ ) ∝ γ a1 −1 e−b1 γ , γ > 0, a1 > 0, b1 > 0, π2 (λ) ∝ λa2 −1 e−b2 λ , λ > 0, a2 > 0, b2 > 0. Here the hyper-parameters a1 , b1 , a2 , b2 are chosen to reflect the prior knowledge about the unknown parameters. A discussion on the selection of the hyper-parameter values is presented in Section 3.2. Now using the joint prior π (γ , λ) = π1 (γ )π2 (λ), the corresponding posterior density given the observed data x = (x(1) , x(2) , . . . , x(m) ) can be written as

π1 (γ )π2 (λ)L(γ , λ | x) , π (γ , λ | x) =  ∞  ∞ π1 (γ )π2 (λ)L(γ , λ | x)dγ dλ 0 0 

∝ γ m+a1 −1 λm+a2 −1 e



− γ b1 +λ b2 +

m 

i=1

 1 x(i)

m 

(1 − zi )γ (Ri +1)−1 .

(5)

i=1

Subsequently, the Bayes estimator of any function, say g (γ , λ) under the squared error loss function, is given by gˆB (γ , λ) = Eγ ,λ|x (g (γ , λ)) =

∞

 0

∞



g (γ , λ)π (γ , λ | x)dγ dλ.

(6)

0

We mention that Dey and Dey [12] have obtained Bayes estimates of unknown parameters using the Lindley’s method (Lindley [25]). Here we propose Metropolis–Hastings (MH) algorithm for computing the desired Bayes estimates. There are two advantages of considering MH algorithm over Lindley’s method. The first advantage is that one need not compute up to third derivatives of the log-likelihood function, and the second advantage is that the samples drawn using MH algorithm can be further utilized to obtain highest posterior density (HPD) intervals for the unknown parameters of the distribution. Notice that the HPD intervals cannot be obtained using Lindley’s method. The algorithm is discussed in the next section.

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3.1. Metropolis–Hastings algorithm Metropolis–Hastings (MH) algorithm is a useful method for generating random samples from the posterior distribution using a proposal density. In general, a symmetric proposal density of type q(θ ′ | θ ) = q(θ | θ ′ ) can be considered, here θ represents vector of unknown parameters of a distribution of interest. We consider a bivariate normal distribution as the proposal density, that is q(θ ′ | θ ) ≡ N2 (θ , Sθ ), here Sθ represents the variance–covariance matrix and θ = (γ , λ). It is to be noted that the bivariate normal distribution may generate negative observations. But in the present work the domain of both shape and scale parameters of the GIE distribution is positive real line. Therefore we propose the following steps of MH algorithm to draw sample from the posterior density given by (5). One may also refer to the work of Dey and Pradhan [14] in this regard. Step 1. Set initial value of θ as θ = θ (0) . Step 2. For i = 1, 2, . . . , M repeat the following steps. (a) Set θ = θ (i−1) . (b) Generate a new candidate parameter value δ from N2 (ln θ , Sθ ). (c) Set θ ′ = exp(δ).   π(θ ′ |x)θ ′

(d) Calculate α = min 1, π(θ |x)θ . (e) Update θ (i) = θ ′ with probability α ; otherwise set θ (i) = θ .

We mention that the choice of Sθ is an important issue in the MH algorithm and the acceptance rate depends upon this. In this regard one may refer to the work of Natzoufras [26] for further details. Finally, from the random sample of size M drawn from the posterior density, some of the initial samples (burn-in) can be discarded, and the remaining samples can be further utilized to compute Bayes estimates. More precisely the expression given by (6) can be evaluated as gˆMH (γ , λ) =

M 

1

M − lB i=l B

g (γi , λi ).

Here lB represents the number of burn-in samples. Next, we propose to use the method of Chen and Shao [9] to compute HPD interval estimates for the unknown parameters of the GIE distribution. The method of Chen and Shao has been extensively used for constructing HPD intervals for the unknown parameters of the distribution of interest. In literature, samples drawn from the posterior density using importance sampling technique are used to construct HPD intervals, see Dey and Dey [12], Kundu and Pradhan [22], and Singh et al. [34]. In the present work, we will utilize the samples drawn using the proposed MH algorithm to construct the interval estimates. More precisely, let us suppose that Π (θ | x) denotes the posterior distribution function of θ . Let us further assume that θ (p) be the pth quantile of θ , that is, θ (p) = inf{θ : Π (θ | x) ≥ p}, where 0 < p < 1. Observe that for a given θ ∗ , a simulation consistent estimator of Π (θ ∗ | x) can be obtained as

Π (θ ∗ | x) =

M 

1

M − lB i=l B

Iθ≤θ ∗ .

Here Iθ≤θ ∗ is the indicator function. Then the corresponding estimate is obtained as

ˆ (θ ∗ | x) = Π

 0   i 

if θ ∗ < θ(lB )

ωj

if θ(i) < θ ∗ < θ(i+1)

   j =l B

if θ ∗ > θ(M )

1

where ωj = by

ˆ (p)

θ

=

1 M −l B

and θ(j) are the ordered values of θj . Now, for i = lB , . . . , M, θ (p) can be approximated

 θ   (lB )

if p = 0

 θ(i)

if

i −1  j=lB

ωj < p <

i  j=lB

ωj .

S. Dey et al. / Statistical Methodology 32 (2016) 185–202

191

  j j+(1−p)M θ ( M ) for j = Now to obtain a 100(1 − p)% HPD credible interval for θ , let Rj =  θ ( M ), lB , . . . , [pM ], here [a] denotes the largest integer less than or equal to a. Then choose Rj∗ among all the R′j s such that it has the smallest width.

3.2. Elicitation of hyper-parameters This section deals with the elicitation of hyper-parameter values when the informative priors are taken into account. Notice that the hyper-parameter values can be selected based on the past available data. In this regard suppose that we have K number of samples available from GIE (γ , λ) distribution, ˆ j ), j = 1, 2, . . . , K . Now on further let the associated maximum likelihood estimates of (γ , λ) be (γˆ j , λ j ˆj equating the mean and variance of (γˆ , λ ), j = 1, 2, . . . , K with the mean and variance of considered priors, the hyper-parameter values can be obtained. In the present work, we have considered the gamma prior π (θ ) ∝ θ h1 −1 e−h2 θ for which Mean(θ ) = h1 /h2 and Variance(θ ) = h1 /h22 . Notice that here θ represents a parameter of interest of the GIE (γ , λ) distribution, and therefore for θ = γ we have h1 = a1 , h2 = b1 , and for θ = λ we have h1 = a2 , h2 = b2 . Therefore, on equating mean and variance of θˆ j , j = 1, 2, . . . , K with the mean and variance of gamma priors, we get K 1 

K j=1

θˆ j =

h1 h2

,

1

and

K  

K − 1 j=1

θˆ j −

K 1 

K i=1

θˆ i

2

h1

=

h22

.

Now on solving the above equations, the estimated hyper-parameters turn out to have the following form

2   K 1 θˆ j K

h1 = 1 K −1

j=1

K   θˆ j − j =1

1 K

2 , K  ˆθ i i =1

1 K

and h2 = 1 K −1

K 

θˆ j

j=1

K   θˆ j − j =1

1 K

2 . K  ˆθ i i=1

One may also refer to a similar type of discussion presented by Dey et al. [13], and Singh and Tripathi [32]. Next we consider the problem of prediction. 4. Point predictor This section deals with the problem of point prediction. Recall that, under progressive type-II censoring at the time of the jth failure x(j) , Rj number of live units are randomly removed from the experiment. Let y(jk) be the lifetimes of those units, k = 1, 2, . . . , Rj and j = 1, 2, . . . , m. Then, we observe that the conditional density of y(jk) given the progressive type-II censored sample x with censoring scheme R, is given by (see Balakrishnan and Cramer [7]) f (y(jk) | x, γ , λ) = k

Rj (F (y(jk) ) − F (x(j) ))k−1 (1 − F (y(jk) ))Rj −k f (y(jk) )

  k

(1 − F (x(j) ))Rj

,

y(jk) > x(j) .

(7)

For the simplicity of notations, we denote f (·; γ , λ) = f (·) and F (·; γ , λ) = F (·). Now the purpose of this section is to predict the y(jk) observations. In subsequent subsections we will discuss different methods of prediction. 4.1. Maximum likelihood predictor In this subsection, we predict the y(jk) observations using maximum likelihood predictor. Observe that the predictive likelihood function (PLF) of y(jk) and (γ , λ), can be written as L1 (y(jk) , γ , λ | x) = f (y(jk) | x, γ , λ)f (x; γ , λ) = f (y(jk) | x, γ , λ)L(γ , λ | x).

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ˆ = v2 (x) are statistics for which In general, if yˆ (jk) = u(x), γˆ = v1 (x) and λ L1 (u(x), v1 (x), v2 (x) | x) =

L1 (y(jk) , γ , λ | x).

sup

(y(jk) ,γ ,λ)

Then u(x) is said to be called maximum likelihood predictor (MLP) of the y(jk) , and v1 (x) and v2 (x) are said to be called the predictive maximum likelihood estimates (PMLEs) of γ and λ, respectively. Therefore, using the predictive log-likelihood function l1 = ln L1 (y(jk) , γ , λ | x), the respective partial derivatives with respect to y(jk) , γ and λ are given by 2 1 (γ (Rj − k + 1) − 1)wjk γ (k − 1)wjk (1 − wjk )γ −1 ∂ l1   = 0, =− + − + ∂ y(jk) λ y(jk) y(jk) (1 − wjk ) y(jk) (1 − zj )γ − (1 − wjk )γ m m   ∂ l1 m+1 Ri ln(1 − zi ) + ln(1 − zi ) = + (Rj − k + 1) ln(1 − wjk ) + ∂γ γ i =1 i=1,i̸=j

(1 − zj )γ ln(1 − zj ) − (1 − wjk )γ ln(1 − wjk ) = 0, (1 − zj )γ − (1 − wjk )γ m m   ∂ l1 zi m+1 1 1 (γ (Rj − k + 1) − 1)wjk Ri = − − + +γ ∂λ λ y(jk) x y ( 1 − w ) x ( 1 − zi ) (jk) jk (i) i=1 (i) i=1,i̸=j + (k − 1)

+ (γ − 1)

zj

m 

zi

i =1

x(i) (1 − zi )

+ γ (k − 1)

x(j)

(1 − zj )γ −1 −

wjk y(jk)

(1 − wjk )γ −1

(1 − zj )γ − (1 − wjk )γ

= 0.

Here wjk = e−λ/y(jk) . Now on solving all the above three equations simultaneously the desired MLP of y(jk) and PMLEs of γ and λ can be obtained. In the present work we use nleqslv package in R-language for solving these equations.

4.2. Best unbiased predictor In this subsection, we consider best unbiased predictor (BUP) to predict y(jk) observations. Notice that a statistic yˆ (jk) used to predict y(jk) is called a BUP of y(jk) , if the predictor error yˆ (jk) − y(jk) has a mean zero, and its prediction error variance Var(ˆyjk − y(jk) ) is less than or equal to that of any other unbiased predictor of y(jk) . Now observe that here the conditional density of y(jk) given the observed data x = (x1 , x2 , . . . , xm ) (see (7)) is just the density of y(jk) given the observed lifetime x(j) . Therefore the BUP of y(jk) is yˆ (jk) = E [y(jk) | x]. Subsequently, using the conditional density, the BUP of y(jk) is ∞



y(jk) f (y(jk) | x, γ , λ)dy(jk) .

yˆ jk = x(j)

Observe that in the conditional density given by (7), put u =

(1−wjk )γ (1−zj )γ

, subsequently, the above

expression reduces to yˆ jk = −

1 Beta(Rj − k + 1, k) (1−w )γ

 0

λ

1

ln(1 −

1

uγ

(1 − zj ))

uRj −k (1 − u)k−1 du.

More precisely, (1−zjk)γ | x ∼ Beta(Rj − k + 1, k) distribution. It is to be further noticed that the shape j and the scale parameters in the above expression are unknown, and they need to be estimated. Thus one would replace both the parameters by their corresponding maximum likelihood estimate values, and subsequently the BUP of y(jk) can be obtained.

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4.3. Conditional median predictor In this subsection, we consider conditional median predictor (CMP) proposed by Raqab and Nagaraja [29] for predicting y(jk) observations. Notice that a predictor yˆ (jk) is called the CMP of y(jk) , if it is the median of the conditional density of y(jk) given the x(j) observation, that is Pγ ,λ y(jk) ≤ yˆ jk | x = Pγ ,λ y(jk) ≥ yˆ jk | x .









(8)

Now observe that the distribution of random variable Therefore the random variable U = 1 −

(1−wjk )γ (1−zj )γ

(1−wjk )γ (1−zj )γ

| x ∼ Beta(Rj − k + 1, k) distribution.

| x ∼ Beta(k, Rj − k + 1) distribution. Subsequently,

using the distribution of U in Eq. (8), we obtain the CMP of y(jk) as yˆ jk =

−λ 1

ln(1 − (1 − M) γ (1 − zj ))

where M stands for median of U ∼ Beta(k, Rj − k + 1) distribution. Further the unknown parameters γ and λ in the above expression can be replaced by their respective maximum likelihood estimates. 4.4. Bayesian predictor In previous subsections, we obtained various predictors under classical approach. In this subsection we consider Bayes predictor to predict the y(jk) observations under a prior π (γ , λ) together with the squared error loss function. Notice that under a prior π (γ , λ) of (γ , λ) the corresponding posterior predictive density is given by f (y(jk) | x) = ∗



∞

0

∞



f (y(jk) | x, γ , λ)π (γ , λ | x)dγ dλ. 0

Subsequently, the kth observation from Rj censored units under the squared error loss can be predicted as ∞



y(jk) f ∗ (y(jk) | x)dy(jk) ,

yˆ (jk) = x(j)

∞



∞  ∞



= 0

x(j)

0

∞



∞



I (γ , λ)π (γ , λ | x)dγ dλ,

= 0



y(jk) f (y(jk) | x, γ , λ)dy π (γ , λ | x)dγ dλ, (9)

0

where I (γ , λ) = −

1 Beta(Rj − k + 1, k)

1



λ 1

0

ln(1 − u γ (1 − zj ))

uRj −k (1 − u)k−1 du.

It is to be noted that the expression given by (9) is nothing but the expectation of I (γ , λ) under the square error loss function. Therefore, we generate samples from the corresponding posterior density using the MH algorithm, subsequently, the desired predictive estimate can be obtained as yˆ (jk) =

1

M 

M − lB i=l B

I (γi , λi ).

5. Prediction interval In previous section, we proposed the various methods to predict y(jk) observations, k = 1, 2, . . . , Rj and j = 1, 2, . . . , m. However, one important aspect may be to construct prediction interval for these observations. In this section we compute various prediction interval estimates.

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5.1. Pivotal method In Section 4.3, we have already discussed that the distribution of random variable (1 − wjk )γ U =1− (1 − zj )γ is Beta(k, Rj − k + 1) distribution. Therefore we can consider U as a pivotal quantity to obtain the prediction interval for y(jk) observations. Subsequently the 100(1 − α)% prediction interval is given by



−λ

,

1



−λ 1

ln(1 − (1 − B α ) γ (1 − zj )) ln(1 − (1 − B1− α ) γ (1 − zj )) 2

.

2

Notice that here Bp represents the 100pth percentile of the Beta(k, Rj − k + 1) distribution and the unknown parameters γ and λ can be replaced by their respective MLEs. 5.2. Highest conditional density method In this section, we propose highest conditional density (HCD) method to construct predictive intervals for y(jk) observations. Observe that the PDF of random variable U is given by (see 4.3) g (u) =

uk−1 (1 − u)Rj −k Beta(k, Rj − k + 1)

,

0
which is unimodal function of u for 1 < k < Rj . Therefore the 100(1 − α)% HCD prediction interval is



−λ

,

1



−λ 1

ln(1 − (1 − w1 ) γ (1 − zj )) ln(1 − (1 − w2 ) γ (1 − zj ))

.

Here, w1 and w2 are the solutions to the following equations w2



w1

g (u)du = 1 − α,

and g (w1 ) = g (w2 ).

More precisely the above equations can be rewritten as Beta(w2 ) (Rj − k + 1, k) − Beta(w1 ) (Rj − k + 1, k) = 1 − α,

and

 1 − w  R j −k 1

1 − w2

Notice that Beta(·) (·, ·) represents the CDF of Beta(·, ·) distribution.

=

 w k−1 2

w1

.

5.3. Bayesian credible interval In this section, we construct 100(1 − α)% Bayesian credible interval estimates. Observe that the survival function corresponding to the conditional density given by (5) is S (t | x, γ , λ) =

P (y(jk) > t | x, γ , λ) P (y(jk) > x(j) | x, γ , λ)

∞ = t∞ x(j)

f (y(jk) | x, γ , λ)dy(jk) f (y(jk) | x, γ , λ)dy(jk)

,

where

   k−1  Rj k−1

∞



f (y(jk) | x, γ , λ)dy(jk) = k t

k

i=0

i

 (−1)k−1−i (1 − zj )γ (i−Rj )

λ

1 − e− t

Rj − i

and



   k−1  Rj k−1

∞

f (y(jk) | x, γ , λ)dy(jk) = k x(j)

k

i=0

i

(−1)k−1−i

1 Rj − i

.

Therefore under a prior π (γ , λ), the associated posterior survival function is given by S ∗ (t | x) =

∞

 0

∞



S (t | x, γ , λ)π (γ , λ | x)dγ dλ. 0

γ (Rj −i) ,

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195

Subsequently, the 100(1 − α)% equal-tail predictive interval (L, U ) is the solution to the following equations S ∗ (L | x) =

α 2

and

S ∗ (U | x) = 1 −

α 2

.

We mention that the sample drawn using the MH algorithm from the posterior density given by (5) can be utilized to solve the above expressions. 5.4. Highest posterior density interval In this section, we construct highest posterior density (HPD) predictive interval for the y(jk) observation. We propose to use the algorithm given by Turkkan and Pham-Gia [36] to obtain the HPD predictive intervals. For the simplicity we reproduce here the algorithm proposed by the authors. Notice that for a given value of α and an input density f (θ ), an (1 − α)% HPD credible interval can be computed using the following algorithm (see, Turkkan and Pham-Gia [36]). Step 1. Start with a value i such that i > f (θ ), ∀ θ .

θ

Step 2. Compute I = θ U f (θ )dθ , while setting f (θ ) = 0 if f (θ ) > i. Here, θL and θU are the left and L right bounds of the range of input density. Step 3. Check if I ≃ α . If not, bisect (0, i) and return to Step 2 until equality is obtained. Step 4. When I = α , compute θi by solving f (θs ) = i, s = 1, 2, . . . . It is to be further  ∞ that in the present work input density will be the posterior predictive density  ∞noticed f ∗ (y(jk) | x) = 0 0 f (y(jk) | x, γ , λ)π (γ , λ | x)dγ dλ. We further mention that the evaluation of the posterior predictive density can be done by generated samples from the posterior density π (γ , λ | x) given by (5) using the MH algorithm. However alternatively the HPD predictive interval (L, U ) can also be obtained by solving the following equations simultaneously U



f ∗ (y(jk) | x)dy(jk) = 1 − α

and f ∗ (L | x) = f ∗ (U | x).

L

6. Data analysis and simulation study In this section, we analyze two real data sets for illustrative purposes and Monte Carlo simulation is performed to compare performance of the proposed methods described in the preceding sections. We used R-statistical programming language for computation. Further, to solve the non-linear equations nleqslv(.) package is used in R-language. 6.1. Data analysis The first data set represents the survival times (in days) of guinea pigs injected with different doses of tubercle bacilli. It is known that guinea pigs have a high susceptibility to human tuberculosis and that is why they were used in this particular study. The regimen number is the common logarithm of the number of bacillary units in 0.5 ml. of challenge solution; i.e., regimen 6.6 corresponds to 4.0 × 106 bacillary units per 0.5 ml is (log(4.0 × 106) = 6.6). Kundu and Howlader [21] used this data to fit the inverse Weibull distribution. Corresponding to regimen 6.6, there were 72 observations listed below:

12, 15, 22, 24, 24, 32, 32, 33, 34, 38, 38, 43, 44, 48, 52, 53, 54, 54, 55, 56, 57, 58, 58, 59, 60, 60, 60, 60, 61, 62, 63, 65, 65, 67, 68, 70, 70, 72, 73, 75, 76, 76, 81, 83, 84, 85, 87, 91, 95, 96, 98, 99, 109, 110, 121, 127, 129, 131, 143, 146, 146, 175, 175, 211, 233, 258, 258, 263, 297, 341, 341, 376.

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Table 1 Goodness-of-fit tests for break down data. PDF WD GD GED GIED

 γ −1 (−x/λ)γ  γ γx e /λ  γ −1 −x/λ  γ x e /λ Γ γ   −x/λ γe (1 − e−x/λ )γ −1 /λ f (x; γ , λ)

γˆ

λˆ

NLC

AIC

BIC

K–S

1.05881

77.5815

58.578

121.156

121.953

0.2166

1.12583

67.3117

58.559

121.118

121.914

0.2217

1.13257

0.01428

58.560

121.120

121.916

0.2213

1.15091

30.2662

58.502

121.005

121.800

0.2107

It may be mentioned that Dube et al. [15] performed the goodness of fit test of the GIE distribution to the above data set and observed that the GIE distribution corresponds to maximum likelihood ˆ = 102.6344 fits the data set very well than exponential, Rayleigh, estimates γˆ = 2.5424 and λ gamma, Weibull and inverse Weibull distributions. As a second application, we consider the data set given in Lawless [23] which represents the break down times of electrical insulating fluid subject to a 30 kV voltage stress. The corresponding break down times (in minutes) are

7.74, 17.05, 20.46, 21.02, 22.66, 43.40, 47.30, 139.07, 144.12, 175.88, 194.90. We first check whether the GIE distribution is suitable for analyzing this data set. We report the MLEs of the parameters and the values of the negative log-likelihood criterion (NLC), Akaike’s information criterion (AIC), Bayesian information criterion (BIC), and Kolmogorov–Smirnov (K–S) test statistic to judge the goodness of fit. The lower the values of these criteria, the better the fit. The parameter estimates and goodness-of-fit statistics are given in Table 1. The reported values suggest that GIE distribution can be considered as an adequate model for the given data set. Therefore, the given data set can be analyzed using this distribution. However, for computational convenience, we further consider the logarithm of the break down times. In this regard, one may also refer to Dey et al. [13]. They observed that the logarithmic data also fits this distribution corresponding to γˆ = 38.761 and λˆ = 14.993. Next, we generate a progressive type-II censored data of size 55 from the first data set corresponding to censoring scheme R = (0∗54 , 17). This censoring scheme represents that, first at 54 observed lifetimes no live unit will be removed, but at 55-th observed lifetime remaining 17 units will be removed from the test. Therefore the generated data set becomes the first 55 observations of the first data set, x(m) = 121, with m = 55. We also generate a progressive type-II censored data of size 7 from the logarithmic data of electrical insulating fluid corresponding to censoring scheme R = (0∗6 , 4). Now for both the generated data sets, we obtain the corresponding maximum likelihood estimates of γ and λ, and their associated 95% asymptotic confidence interval estimates. We also compute Bayes estimates using the MH algorithm under the non-informative prior. Note that the non-informative prior, π (γ , λ) = 1/(γ λ) becomes the particular case of the proposed prior corresponding to hyper-parameter values a1 = b1 = a2 = b2 = 0. It is to be noted that, while generating samples from the posterior distribution using the MH algorithm, initial values of (γ , λ) are ˆ . Furthermore, we considered the variance–covariance matrix Sθ considered as (γ (0) , λ(0) ) = (γˆ , λ) ˆ of (ln γˆ , ln λ), which can be easily obtained using the delta method. Finally, we discarded 1000 burnin samples among the total 6000 samples generated from the posterior density, and subsequently obtained Bayes estimates, and HPD interval estimates using the method of Chen and Shao [9]. In Table 2 all the estimated values of maximum likelihood, Bayesian and associated interval estimates for both the real data sets are presented. From Table 2, it is observed that MLEs compete well with non-informative Bayes estimates. Next, we consider the problem of prediction. We predict the first 4 censored observations under the proposed classical and Bayesian approaches. All the predictive observations are reported in Table 3. Further, we present the associated predictive interval estimates in Table 4. We observe that the predictive estimated values are smaller in case of MLP followed by CMP, BUP, and Bayes predictor. In both Tables 3 and 4, we have reported the true observations of the considered data sets. It is observed that the predictive estimates using Bayes predictor are more closer to the true observations than

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197

Table 2 Maximum likelihood, Bayesian and associated interval estimates for real data set. Data set

Parameter

Maximum likelihood estimation

Bayesian estimation

MLE

Asymptotic confidence interval

MH algorithm

HPD interval

1

γ λ

2.4939 101.336

(1.2675, 3.7203) (71.651, 131.022)

2.5008 101.465

(1.8383, 3.1864) (98.746, 104.151)

2

γ λ

62.704 16.081

(0, 256.52) (5.1685, 29.994)

75.019 16.451

(46.905, 106.438) (12.951, 19.877)

Table 3 Predictive estimates using various predictors. Data set

j

k

True value

PMLEs

1

55

1 2 3 4

127 129 131 143

γˆ γˆ γˆ γˆ

2

7

1 2 3 4

4.9349 4.9706 5.1698 5.2724

–

yˆ (jk) MLP

BUP

CMP

Bayes

= 2.595, λˆ = 102.897 = 2.619, λˆ = 103.547 = 2.623, λˆ = 103.635 = 2.628, λˆ = 103.728

122.891 125.385 130.125 135.276

125.556 130.501 135.901 141.835

124.116 128.860 134.079 139.812

125.631 130.645 136.232 142.301

γˆ = 136.255, λˆ = 18.488 γˆ = 141.392, λˆ = 18.670 γˆ = 150.758, λˆ = 18.943

– 4.0692 4.3228 4.6845

4.0682 4.3215 4.6588 5.2387

4.0123 4.2667 4.5990 5.1444

4.0815 4.3654 4.7116 5.2767

Note: – : no suitable solution found. Table 4 Predictive interval estimates. Data set

j

k

True value

Pivotal

HCD

Credible

HPD

1

55

1 2 3 4

127 129 131 143

(121.11, 138.09) (122.12, 148.20) (123.96, 158.22) (126.42, 168.80)

(121.00, 134.78) (121.26, 144.14) (122.76, 154.10) (125.38, 163.65)

(121.11, 138.65) (122.05, 149.31) (123.92, 160.29) (126.32, 171.79)

(121.00, 135.18) (121.15, 144.77) (122.24, 154.97) (124.06, 165.71)

2

7

1 2 3 4

4.9349 4.9706 5.1698 5.2724

(3.8624, 4.5785) (3.9210, 5.0291) (4.0487, 5.6088) (4.2816, 6.7357)

(3.8563, 3.9599) (3.8981, 4.9839) (4.0844, 5.6290) –

(3.8623, 4.6961) (3.9234, 5.1880) (4.0500, 5.7889) (4.2660, 6.9101)

(3.8565, 4.5689) (3.8715, 5.0168) (3.9657, 5.6013) (4.1349, 6.6330)

Note: – : no suitable solution found.

the other predictors. All the prediction intervals contain the true observations except for the first observation in second data set, and it is found that prediction intervals obtained using HCD method have smaller interval lengths as compared to the pivotal method. In a similar way, under Bayesian approach, HPD prediction intervals have smaller interval length as compared to equal-tail predictive intervals. It is also observed that for the higher predictive observations, the length of associated predictive intervals and PMLEs does increase. 6.2. Simulation study In this section, we conduct simulation study to compare the performance of proposed method of estimation and prediction. We simulate data from GIE (0.5, 1) distribution under various progressive type-II censoring schemes for different choices of (n, m). Based on the generated data, we first compute maximum likelihood estimates and associated 95% asymptotic confidence interval estimates. Note that the initial guess values are considered to be same as the true parameter values while obtaining maximum likelihood estimates. Further, we compute Bayes estimates using the MH algorithm. While employing MH algorithm, we take into account the maximum likelihood estimates ˆ . These estimates as initial guess values, and the associated variance–covariance matrix of (ln γˆ , ln λ)

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Table 5 Average estimated values, interval estimates, MSEs, AILs and CPs. n

20

m

8

R

MLE

(12, 0∗7 )

γ λ

(0 , 12) ∗7

γ λ

10

(10, 0∗9 )

γ λ

(0∗9 , 10)

γ λ

25

10

(15, 0 ) ∗9

γ λ

(0∗9 , 15)

γ λ

15

(10, 0∗14 )

γ λ

(0

∗14

, 10)

γ λ

P-I

P-II

Avg MSE

Asy CI AIL/CP

Avg MSE

HPD interval AIL/CP

Avg MSE

HPD interval AIL/CP

0.6473 0.1206 1.2111 0.3087 0.8975 0.4109 1.3969 0.3935 0.6082 0.0667 1.2056 0.2675 0.7233 0.2407 1.3049 0.3629 0.5832 0.0597 1.1512 0.2107 0.6670 0.1490 1.2071 0.2071 0.5806 0.0381 1.1589 0.1724 0.6096 0.0676 1.1858 0.2101

(0.0965, 1.1982) 1.1017/96.5 (0.2549, 2.1719) 1.9170/96.2 (0.0034, 2.0696) 2.0662/96.7 (0.2282, 2.5627) 2.3345/96.1 (0.1514, 1.0649) 0.9135/96.7 (0.2725, 2.1402) 1.8677/96.8 (0.0418, 1.4527) 1.4109/96.9 (0.2720, 2.3421) 2.0701/95.9 (0.1490, 1.0173) 0.8683/95.7 (0.3038, 2.0032) 1.6994/95.9 (0.0222, 1.3601) 1.3379/97.9 (0.2687, 2.1444) 1.8757/99.1 (0.2280, 0.9332) 0.7052/97.6 (0.3697, 1.9499) 1.5802/97.7 (0.1690, 1.0502) 0.8812/96.8 (0.3705, 2.0017) 1.6312/97.1

0.6327 0.1080 1.2163 0.2742 0.9573 0.4605 1.3606 0.3743 0.5942 0.0591 1.2007 0.2325 0.735 0.2634 1.2734 0.3809 0.5712 0.0538 1.1562 0.1923 0.6836 0.1649 1.1756 0.1589 0.5683 0.0343 1.1449 0.1535 0.6018 0.0642 1.1550 0.1755

(0.1833, 1.1919) 1.0086/97.1 (0.4235, 2.1516) 1.7281/97.2 (0.1585, 2.3337) 2.1752/98.9 (0.4499, 2.4256) 1.9757/94.8 (0.2037, 1.0630) 0.8593/97.1 (0.4279, 2.1099) 1.6820/96.8 (0.1793, 1.5270) 1.3477/97.8 (0.4417, 2.2315) 1.7898/95.1 (0.1973, 1.0197) 0.8224/95.8 (0.4360, 1.9955) 1.5595/97.1 (0.1589, 1.4603) 1.3014/98.8 (0.4076, 2.0374) 1.6298/99.7 (0.2490, 0.9347) 0.6857/98.2 (0.4582, 1.9157) 1.4575/97.9 (0.2243, 1.0656) 0.8413/97.8 (0.4575, 1.9286) 1.4711/96.3

0.5154 0.0026 1.0694 0.0262 0.5198 0.0021 1.0622 0.0234 0.5161 0.0028 1.0744 0.0265 0.5174 0.0025 1.0645 0.0260 0.5111 0.0027 1.0681 0.0261 0.5121 0.0019 1.0561 0.0187 0.5195 0.0031 1.0709 0.0248 0.5181 0.0029 1.0637 0.0236

(0.3073, 0.7442) 0.4369/98.3 (0.5894, 1.5987) 1.0093/98.6 (0.3069, 0.7562) 0.4493/99.2 (0.6095, 1.5535) 0.9440/97.5 (0.3144, 0.7369) 0.4225/98.7 (0.5978, 1.5988) 1.001/96.9 (0.3113, 0.7450) 0.4337/98.2 (0.6086, 1.5616) 0.9530/96.9 (0.3110, 0.7313) 0.4203/97.3 (0.6027, 1.5780) 0.9753/98.6 (0.3045, 0.7426) 0.4381/99.5 (0.6212, 1.5310) 0.9098/99.7 (0.3307, 0.7250) 0.3943/98.9 (0.6188, 1.5641) 0.9453/99.1 (0.3248, 0.7297) 0.4049/98.7 (0.6270, 1.5343) 0.9073/98.4

Note: Asy CI—asymptotic confidence interval, Avg—average.

are obtained under both the non-informative and informative priors. For informative priors, we generate 100 complete samples each of size 30 from GIE (0.5, 1) distribution as past samples, and subsequently obtain the hyper-parameter values as a1 = 18.09, b1 = 35.26, a2 = 10.93, b2 = 10.03. These values are then plugged-in to compute the desired estimates. All the average estimates are reported in Table 5. Note that in Table 5, non-informative and informative priors are respectively denoted as P-I and P-II. Further, the first row represents the average estimates and interval estimates, and in the second row, associated means square errors (MSEs) and average interval length (AILs) with coverage percentages (CPs) are reported. From tabulated values it is observed that based on MSEs, higher values of n and m lead to better estimates. It is also observed that non-informative Bayes estimates compete well with the maximum likelihood estimates, and the performance of the Bayes estimates obtained under informative prior is appreciable. It can also be seen that under informative prior the AILs and associated CPs of HPD intervals are better than those of non-informative priors. Furthermore, it is also seen that MSEs, and AILs of associated interval estimates are generally higher when the units are removed at later stage. Next based on the generated data we predict the first two observations corresponding to the considered censoring schemes. All the average estimated values are reported in Table 6. From tabulated values it is seen that the predictive observations using MLP are generally smaller than CMP followed by BUP and Bayes predictors. It is also seen that with higher predictive observations, PMLEs do increase. Further, Bayes predictor under informative prior has smaller estimated values than that of non-informative prior. Here all the predicted observations lie between the corresponding predictive

S. Dey et al. / Statistical Methodology 32 (2016) 185–202

199

Table 6 Average values of predictive estimates. n

m

R

j

k

γˆ 20

8

10

(12, 0∗7 )

1

(0∗7 , 12)

8

(10, 0∗9 )

1

(0 , 10)

10

(15, 0∗9 )

1

(0 , 15)

10

(10, 0∗14 )

1

(0∗14 , 10)

15

∗9

25

10

∗9

15

yˆ (jk)

PMLEs

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

λˆ

0.6467 0.6915 0.6822 0.7182 0.6101 0.6833 0.6438 0.7059 0.5942 0.6428 0.6648 0.7201 0.5581 0.6186 0.6056 0.6697

MLP

1.1579 1.2177 1.1766 1.2502 1.1482 1.2079 1.1552 1.2429 1.1569 1.2115 1.1469 1.2080 1.1475 1.2302 1.1947 1.2631

BUP

0.5792 0.6330 2.2515 2.6087 0.5954 0.6819 3.7120 4.1151 0.4831 0.5577 1.6014 2.4373 0.5455 0.6755 5.6918 7.0371

0.6876 1.0104 2.8052 3.6531 0.7470 1.2136 4.6514 6.1236 0.6061 0.8397 2.6972 3.0805 0.7436 1.1818 7.4948 10.276

CMP

0.6029 0.8754 2.5806 3.2324 0.6346 1.0075 4.1798 5.2309 0.5405 0.7529 2.5307 2.8520 0.6301 0.9842 6.7799 8.8285

Bayes predictor P-I

P-II

0.7523 1.1770 2.8591 3.7880 0.8491 1.4396 4.6421 6.6367 0.6512 0.9466 2.7601 3.4225 0.7951 1.3462 7.7567 10.947

0.7120 1.0772 2.8568 3.7388 0.7728 1.2782 4.6865 6.2145 0.6213 0.8815 2.7286 3.2059 0.7579 1.2298 7.5339 10.328

Table 7 Average interval lengths of predictive interval estimates. n

20

m

8

10

25

10

15

R

j

k

(12, 0∗7 )

1

(0∗7 , 12)

8

(10, 0 )

1

(0∗9 , 10)

10

(15, 0∗9 )

1

(0∗9 , 15)

10

(10, 0∗14 )

1

∗9

(0

∗14

, 10)

15

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

Pivotal

1.0577 1.8643 2.3935 5.0101 1.3540 2.6134 4.9204 10.159 0.8542 1.3257 1.8233 2.9571 1.3931 2.5859 7.5199 16.648

HCD

0.8452 1.5514 1.8206 3.9529 1.0611 2.1250 3.6465 7.8401 0.6925 1.1391 1.4056 2.4238 1.0942 2.1104 5.6072 12.904

P-I

P-II

ET

HPD

ET

HPD

1.4933 3.0093 3.0429 6.9765 1.8851 4.1699 5.8928 14.686 1.1064 2.0271 2.2979 4.7954 1.7543 3.8174 9.3934 22.985

1.1294 2.3725 2.1512 4.8287 1.4009 3.3745 4.0929 9.9253 0.8611 1.6460 1.6699 3.4740 1.3202 3.3304 6.6069 15.781

1.2622 2.4631 2.7254 5.7458 1.5717 3.3308 5.1994 11.201 0.9861 1.7655 2.0212 3.7201 1.5517 3.2172 7.8995 17.961

0.9733 2.2651 2.0280 4.2514 1.1940 2.8768 3.7988 8.1286 0.7770 1.5117 1.5314 2.8583 1.1871 3.3317 5.7922 13.078

Note: ET–equal-tail.

intervals, and to compare the performance of the various prediction intervals we report AILs in Table 7. It is observed that intervals obtained using the HCD method have smaller AILs as compared to the pivotal method. In a similar way, HPD prediction intervals have smaller AILs as compared to equal-tail prediction intervals. Finally AILs of prediction intervals obtained under informative prior are smaller than those obtained under non-informative prior. 7. Conclusion In this paper, we have considered the problem of estimation and prediction for generalized inverted exponential distribution under progressive type-II censoring from classical and Bayesian viewpoints. We derived maximum likelihood estimates and associated asymptotic confidence interval estimates for the unknown parameters of a GIE distribution. We then computed Bayes estimates and the corresponding HPD interval estimates under non-informative and informative priors. We have also discussed as to how to select the values of hyper-parameters based on past samples when

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S. Dey et al. / Statistical Methodology 32 (2016) 185–202

informative prior is taken into account. Our simulation study reveals that MLEs compete well with non-informative Bayes estimates, and the performance of estimates under informative prior is better than the non-informative prior. As regard to prediction, we predicted censored observations using maximum likelihood predictor, best unbiased predictor, conditional median predictor and Bayesian predictor. We also computed the associated predictive interval estimates using pivotal method, highest conditional density method, and credible and HPD intervals under Bayesian setup. It is found that the predictive intervals contain the true observations of the given data set, and this observation holds true for both the classical and Bayesian approaches. We have also observed that predictive intervals obtained using highest conditional method and highest posterior density interval have smaller interval length respectively as compared to pivotal method and equal-tail interval. Though we have considered squared error loss function under Bayesian set up, yet other loss functions can also be considered. The present work can also be extended to other censoring schemes such as hybrid censoring and progressive first-failure censoring. Acknowledgments The authors wish to express their sincere gratitude to an anonymous reviewer for several constructive suggestions which helped in restructuring introduction and simulation sections. They also thank the Editor and an Associate Editor for their encouraging comments. The author Yogesh Mani Tripathi gratefully acknowledges the partial financial support from the Department of Science and Technology, India, under the grant SR/S4/MS: 785/12. Appendix Proof of Lemma 1. It is known that if X(1) < X(2) < · · · < X(m) is a progressive type-II censored sample from a GIE(γ , λ) distribution, then the pdf of X(i) is fX(i) (x) = ci−1

i 

ai,k f (x)[1 − F (x)]ηk −1

k =1

= γ λci−1

i 

ai , k

e−λ/x x2

k=1

(1 − e−λ/x )γ ηk −1

for x > 0 and 0 otherwise (see for example, Balakrishnan and Aggarwala [6, p. 26.]). Then

 E

e−λ/X(i)



X(i) (1 − e−λ/X(i) )

∞

 =

e−λ/xi

= γ λci−1

i 

=

γ ci−1 λ

ci−1

i 

λ

k=1

∞

 ai,k

e−2λ/xi x3i

0

k=1

=−

fX(i) (xi )dxi

xi (1 − e−λ/xi )

0

i 

ai,k



y ln(y)(1 − y)γ ηk −2 dy

0

k=1

 ai , k

1



(1 − e−λ/xi )γ ηk −2 dxi

 ψ(γ ηk + 1) − ψ(2) . ηk (γ ηk − 1)

Similarly,

 E

e−λ/X(i) X(2i) (1 − e−λ/X(i) )2



∞

 = 0

e−λ/xi x2i (1 − e−λ/xi )2

= γ λci−1

i  k=1

∞

 ai , k 0

fX(i) (xi )dxi

e−2λ/xi x4i

(1 − e−λ/xi )γ ηk −3 dxi

S. Dey et al. / Statistical Methodology 32 (2016) 185–202

=

=

i γ ci−1  ai , k λ2 k=1

1



201



y[ln(y)]2 (1 − y)γ ηk −3 dy

0

i  γ c ai,k [(γ ηk − 1)(γ ηk − 2)[ψ1 (2) − ψ1 (γ ηk ) i − 1 λ2 k=1  + (ψ(2) − ψ(γ ηk ))2 ] .

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