Estimation and Prediction for the Poisson-Exponential Distribution Based on Type-II Censored Data

SYNOPTIC ABSTRACT

This article addresses the problems of estimation and prediction when the lifetime data following Poisson-Exponential distribution are observed under type-II censoring. We obtain maximum likelihood estimates and associated interval estimates under a classical approach, and Bayes estimates using various loss functions and associated highest posterior density interval estimates. Maximum likelihood estimates are obtained using the Newton-Raphson method and Expectation Maximization (EM) algorithm, and Bayes estimates are computed using importance sampling and Lindley approximation. We also compute shrinkage preliminary test estimates based on maximum likelihood and Bayes estimates. Further, we provide inference on the censored observations by making use of best unbiased and condition median predictors under a classical approach, and predictive estimates under the Bayesian paradigm using importance sampling. The associated predictive interval estimates are also obtained using different methods. Finally, we conduct a simulation study to compare the performance of all the proposed methods of estimation and prediction, and analyze a real data set for illustration purpose.

KEY WORDS AND PHRASES:

• Bayesian estimation
• Poisson-exponential distribution
• EM algorithm
• shrinkage estimation
• Lindley approximation
• prediction
• Type-II censoring

1. Introduction

The exponential distribution is the simplest and the most widely used lifetime model in life testing and reliability analysis. However, the main drawback of this distribution is that it may not be suitable for modeling the failure phenomena of devices whose failure rate are not constant over time. Recently, Cancho et al. (2011) proposed Poisson exponential (PE) distribution, which can accommodate data with increasing failure rate. The traditional exponential distribution can also be seen as a particular case of this distribution. The main advantage of this distribution is that its genesis is based on complementary risk problems in the presence of latent risks. In complementary risk problems, the maximum lifetime value among all risks is considered as lifetime of a component rather than associated with a particular risk, whereas in latent risks, the information about the true cause of failure may not be available. For an example, a system in an experiment may contain many components and the cause of failure of the system may be due to failure of any component. Such types of problems often arise in the field of life testing and reliability analysis (see Basu & Klein, 1982). The PE distribution can be derived in the following way. Suppose that random variables M and T_j, respectively, denote the number of competing risks and time to the occurrence of an event of interest due to the j-th competing risk, $j=1,2 \dots$. Further, assume that M has a truncated Poisson distribution with probability mass function given by $P(M=m)=e^{-\theta }{\theta }^m/m(1-e^{-\theta }),m=1,2,\dots ,\theta >0$, and given M = m the random variables $T_j,j=1,2 \dots ,m$ are independent and identically distributed by exponential density function with rate parameter λ. Then, the maximum value of all causes, say random variable $X=\text{max}(T_1,T_2,\dots ,T_M)$, follows Poisson exponential distribution, PE$(\theta ,\lambda )$, with probability density function (PDF) and cumulative distribution function (CDF) of the following form. (1) $$f(x;\theta ,\lambda )=\frac{\theta \lambda e^{-\lambda x-\theta e^{-\lambda x}}}{1-e^{-\theta }},x>0,\theta >0,\lambda >0,$$(1) (2) $$F(x;\theta ,\lambda )=1-\frac{1-e^{-\theta e^{-\lambda x}}}{1-e^{-\theta }},$$(2) where θ is the shape parameter, and λ is the scale parameter that also denotes the failure rate of the distribution of time-to-event due to individual complementary risks. Further, $\theta /(1-e^{-\theta })$ represents the mean of the number of complementary risks. Notice that this distribution converges to an exponential distribution with parameter λ when θ approaches zero.

One may refer to Cancho et al. (2011) for a formal proof of its PDF, mean, variance, explicit algebraic formulae for its reliability, and failure rate functions, quantiles and moments. In their work, the authors also proposed the EM algorithm to compute maximum likelihood estimates, and further obtained Fisher information matrix to compute the asymptotic variance-covariance matrix. Later, Louzada-Neto et al. (2011) discussed statistical inference for the distribution using non-informative prior under a Bayesian approach. The authors also presented a discussion on model selection followed by the analysis on a real data set using developed methodology. Singh et al. (2014) also considered the problem of inference under a Bayesian approach and computed Bayes estimates using symmetric and asymmetric loss functions, and associated highest posterior density interval estimates. One may further refer to the work of Rodrigues et al. (2016) for different methods of estimation. So far, all the work on the PE distribution have been based on complete samples, and this distribution has not received much attention under type-II censored data which occurs due to time restrictions and cost constraints. This motivates us to write this article with two main objectives. The first objective is to provide the statistical inference about the unknown parameters of the PE distribution when the lifetime data are observed under type-II censoring. The second objective is to provide the statistical inference about the censored observations. We consider both problems of estimation and prediction under classical as well as Bayesian approaches. Further, Bayesian shrinkage estimates and predictive interval estimates are also constructed. The rest of this article is organized as follows. Section 2 deals with maximum likelihood estimation, and Section 3 discusses Bayesian estimation. Section 4 discusses shrinkage preliminary test estimators. In Section 5, the problem of prediction is discussed. Real data set and the simulation study are, respectively, discussed in Sections 6 and 7. Finally, a conclusion is presented in Section 8.

2. Maximum Likelihood Estimation

This section deals with obtaining maximum likelihood estimators for the unknown parameters of PE distribution when the lifetime data are observed under type-II censoring. Suppose that n number of units whose lifetimes, say $\mathit{W}=(W_1,W_2,\dots ,W_n)$ follow PE distribution, are put on life test experiment, and the experiment terminates after a prefixed $r(\le n)$ number of units fail. Now, let us assume that $\mathit{x}=(x_1,x_2,\dots ,x_r)$ represent the lifetimes of observed failures. Then, the likelihood function of $(\theta ,\lambda )$ given the observed data $\mathit{x}$ can be written as, (3) $$L(\theta ,\lambda )=C\prod _{i=1}^{r}f(x_{\left(i\right)};\theta ,\lambda ){[1-F(x_{\left(r\right)};\theta ,\lambda \left)\right]}^{n-r},$$(3) where $C=n!/(n-r)!$, and PDF and CDF are, respectively, given by (1)(3) $$L(\theta ,\lambda )=C\prod _{i=1}^{r}f(x_{\left(i\right)};\theta ,\lambda ){[1-F(x_{\left(r\right)};\theta ,\lambda \left)\right]}^{n-r},$$(3) and (2)(2) $$F(x;\theta ,\lambda )=1-\frac{1-e^{-\theta e^{-\lambda x}}}{1-e^{-\theta }},$$(2) . Let $l=\hspace{0.17em}\text{ln}\hspace{0.17em}L(\theta ,\lambda )$, be the log-likelihood function, then the maximum likelihood estimates of θ and λ can be obtained on simultaneously solving the partial differential equations of the l with respect to θ and λ and on equating them to zero. We first compute maximum likelihood estimates using the Newton-Raphson (NR) method, and for comparison purposes, we employ the Expectation-Maximization (EM) algorithm proposed by Dempster et al. (1977). This algorithm is very useful for the situations, particularly, when the data are censored in nature. The algorithm is discussed in the next section.

2.1. EM Algorithm

EM algorithm starts with writing likelihood function given the complete sample W. However, we have observed data $\mathit{x}=(x_1,x_2,\dots ,x_r)$ under type-II censoring, and therefore, we do not have lifetimes of n – r censored observations. Now, let us assume that the lifetimes of the censored observations are $z=(z_1,z_2,\dots ,z_{n-r})$. Therefore, the complete sample W can be seen as a combination of observed data X and censored data Z, that is $W=(X,Z)$. Subsequently, on equating the partial derivatives of the complete log-likelihood function with respect to $(\theta ,\lambda )$ to zeros, we get, (4) $$\frac{\partial l}{\partial \theta }=\frac{n}{\theta }-\frac{n}{e^{\theta }-1}-(\sum _{i=1}^r{e}^{-\lambda x_i}+\sum _{i=1}^{n-r}{e}^{-\lambda z_i})=0,$$(4) (5) $$\frac{\partial l}{\partial \lambda }=\frac{n}{\lambda }-(\sum _{i=1}^r{x}_i+\sum _{i=1}^{n-r}{z}_i)+\theta (\sum _{i=1}^r{x}_i{e}^{-\lambda x_i}+\sum _{i=1}^{n-r}{z}_i{e}^{-\lambda z_i})=0,$$(5)

Now, the first step in the EM algorithm is the Expectation (E-) step in which the censored observations are replaced with the expected observations, and the second step is the Maximization (M-) step which looks for the estimated values of $(\theta ,\lambda )$, called maximum likelihood estimates, that maximizes the E-step. Suppose that $({\theta }^{(k)},{\lambda }^{(k)})$ are the estimates of $(\theta ,\lambda )$ at the k-th stage, then the (k + 1)-th stage estimate of λ can be obtained upon solving Equation (6)(6) $$\frac{n}{{\theta }^{\left(k\right)}}-\frac{n}{e^{{\theta }^{\left(k\right)}}-1}-(\sum _{i=1}^r{e}^{-{\lambda }^{(k+1)}{x}_i}+(n-r\left)E_1\right({\theta }^{\left(k\right)},{\lambda }^{\left(k\right)}\left)\right) = 0,$$(6) (using Equation (4)(4) $$\frac{\partial l}{\partial \theta }=\frac{n}{\theta }-\frac{n}{e^{\theta }-1}-(\sum _{i=1}^r{e}^{-\lambda x_i}+\sum _{i=1}^{n-r}{e}^{-\lambda z_i})=0,$$(4) ), and further given the value of ${\lambda }^{(k+1)}$ the k + 1-th stage estimator of θ is given by (7)(7) $${\theta }^{(k+1)} = \frac{({\sum }_{i=1}^r{x}_i+(n-r\left)E_2\right({\theta }^{\left(k\right)},{\lambda }^{\left(k\right)}\left)\right)-n{\left({\lambda }^{(k+1)}\right)}^{-1}}{({\sum }_{i=1}^r{x}_i{e}^{-{\lambda }^{\left(k\right)}{x}_i}+(n-r\left)E_3\right({\theta }^{\left(k\right)},{\lambda }^{\left(k\right)}\left)\right)},$$(7) (see Equation (5)(5) $$\frac{\partial l}{\partial \lambda }=\frac{n}{\lambda }-(\sum _{i=1}^r{x}_i+\sum _{i=1}^{n-r}{z}_i)+\theta (\sum _{i=1}^r{x}_i{e}^{-\lambda x_i}+\sum _{i=1}^{n-r}{z}_i{e}^{-\lambda z_i})=0,$$(5) ). (6) $$\frac{n}{{\theta }^{\left(k\right)}}-\frac{n}{e^{{\theta }^{\left(k\right)}}-1}-(\sum _{i=1}^r{e}^{-{\lambda }^{(k+1)}{x}_i}+(n-r\left)E_1\right({\theta }^{\left(k\right)},{\lambda }^{\left(k\right)}\left)\right) = 0,$$(6) (7) $${\theta }^{(k+1)} = \frac{({\sum }_{i=1}^r{x}_i+(n-r\left)E_2\right({\theta }^{\left(k\right)},{\lambda }^{\left(k\right)}\left)\right)-n{\left({\lambda }^{(k+1)}\right)}^{-1}}{({\sum }_{i=1}^r{x}_i{e}^{-{\lambda }^{\left(k\right)}{x}_i}+(n-r\left)E_3\right({\theta }^{\left(k\right)},{\lambda }^{\left(k\right)}\left)\right)},$$(7) where, $$\begin{array}{c}{E}_1(\theta ,\lambda )=E\left(e^{-\lambda Z_i}\right|Z_i>X_r)=\frac{\theta \lambda {(1-e^{-\theta })}^{-1}}{1-F(x_r;\theta ,\lambda )}{\int }_{x_r}^{\infty }{e}^{-2\lambda x-\theta e^{-\lambda x}}dx,\\ E_2(\theta ,\lambda )=E\left(Z_i\right|Z_i>X_r)=\frac{\theta \lambda {(1-e^{-\theta })}^{-1}}{1-F(x_r;\theta ,\lambda )}{\int }_{x_r}^{\infty }x{e}^{-\lambda x-\theta e^{-\lambda x}}dx,\end{array}$$ and $$E_3(\theta ,\lambda )=E\left(Z_i{e}^{-\lambda Z_i}\right|Z_i>X_r) = \frac{\lambda \theta {(1-e^{-\theta })}^{-1}}{1-F(x_r;\theta ,\lambda )}{\int }_{x_r}^{\infty }x{e}^{-2\lambda x-\theta e^{-\lambda x}}dx,$$

The iterative procedure in Equations (6)(6) $$\frac{n}{{\theta }^{\left(k\right)}}-\frac{n}{e^{{\theta }^{\left(k\right)}}-1}-(\sum _{i=1}^r{e}^{-{\lambda }^{(k+1)}{x}_i}+(n-r\left)E_1\right({\theta }^{\left(k\right)},{\lambda }^{\left(k\right)}\left)\right) = 0,$$(6) and (7)(7) $${\theta }^{(k+1)} = \frac{({\sum }_{i=1}^r{x}_i+(n-r\left)E_2\right({\theta }^{\left(k\right)},{\lambda }^{\left(k\right)}\left)\right)-n{\left({\lambda }^{(k+1)}\right)}^{-1}}{({\sum }_{i=1}^r{x}_i{e}^{-{\lambda }^{\left(k\right)}{x}_i}+(n-r\left)E_3\right({\theta }^{\left(k\right)},{\lambda }^{\left(k\right)}\left)\right)},$$(7) can be terminated on achieving a convergence, that is, $|{\theta }^{(k+1)}-{\theta }^{(k)}|+|{\lambda }^{(k+1)}-{\lambda }^{(k)}|<ϵ$ for a small value of ϵ. For uniqueness and existence of maximum likelihood estimates, one may refer to the discussion presented by Cancho et al. (2011). Next, we compute interval estimates for the unknown parameters of the PE distribution.

2.2. Fisher Information Matrix

This section deals with obtaining the Fisher information matrix, which will be further used to compute interval estimates. Notice, that another advantage of employing the EM algorithm is the computation of the Fisher information matrix using the idea of missing information principle suggested by Louis (1982). The idea suggests that Observed information = Complete information − Missing information. Subsequently, it can be expressed in the following way. (8) $$I_X(\theta ,\lambda )=I_W(\theta ,\lambda )-I_{W|X}(\theta ,\lambda )$$(8)

Now, the complete information $I_W(\lambda ,\theta )$ can be obtained using the complete sample, and is given by, $$I_W(\lambda ,\theta )=\left[\begin{array}{cc}\frac{n}{{\lambda }^2}+n\lambda {\theta }^2{(1-e^{-\theta })}^{-1}{\int }_0^{\infty }{x}^2{e}^{-2\lambda x-\theta e^{-\lambda x}}dx& n\lambda \theta {(1-e^{-\theta })}^{-1}{\int }_0^{\infty }x{e}^{-2\lambda x-\theta e^{-\lambda x}}dx\\ n\lambda \theta {(1-e^{-\theta })}^{-1}{\int }_0^{\infty }x{e}^{-2\lambda x-\theta e^{-\lambda x}}dx& \frac{n}{{\theta }^2}-\frac{n{e}^{-\theta }}{{(1-e^{-\theta })}^2}\end{array}\right].$$

Further, the missing information can be computed as, $$I_{W|X}(\theta ,\lambda ) =(n-r)I_{Z_i|X}(\theta ,\lambda ) = (n-r)\left[\begin{array}{cc}{a}_{11}(x_r,\theta ,\lambda )& a_{12}(x_r,\theta ,\lambda )\\ a_{21}(x_r,\theta ,\lambda )& a_{22}(x_r,\theta ,\lambda )\end{array}\right]$$ where the expression in the above matrix are obtained using the conditional density $f_{Z|X}(Z_i|X=x_r)=f(z_i;\theta ,\lambda )/(1-F(x_r;\theta ,\lambda )),z_i>x_r$, and are given by, $$\begin{array}{c}{a}_{11}(x,\theta ,\lambda )=\frac{1}{{\lambda }^2}-\frac{\theta x^2{e}^{-\theta e^{-\lambda x-\lambda x}}[e^{-\theta e^{-\lambda x}}+\theta e^{-\lambda x}-1]}{{(1-e^{-\theta e^{-\lambda x}})}^2},\\ a_{12}(x,\theta ,\lambda )=a_{21}(x,\theta ,\lambda )=-\frac{e^{-\theta e^{-\lambda x-\lambda x}}[1-e^{-\theta e^{-\lambda x}}-\theta e^{-\lambda x}]}{{(1-e^{-\theta e^{-\lambda x}})}^2}\\ -\theta \lambda {(1-e^{-\theta })}^{-1}{\int }_0^{\infty }x{e}^{-2\lambda x-\theta e^{-\lambda x}}dx,\\ a_{22}(x,\theta ,\lambda )=\frac{1}{{\theta }^2}-\frac{e^{-\theta e^{-\lambda x}-2\lambda x}}{{(1-e^{-\theta e^{-\lambda x}})}^2}.\end{array}$$

Now, the asymptotic variance-covariance matrix of $(\widehat{\theta },\widehat{\lambda })$ can be obtained by inverting the Fisher information matrix given by (8)$$\begin{array}{c}{E}_1(\theta ,\lambda )=E\left(e^{-\lambda Z_i}\right|Z_i>X_r)=\frac{\theta \lambda {(1-e^{-\theta })}^{-1}}{1-F(x_r;\theta ,\lambda )}{\int }_{x_r}^{\infty }{e}^{-2\lambda x-\theta e^{-\lambda x}}dx,\\ E_2(\theta ,\lambda )=E\left(Z_i\right|Z_i>X_r)=\frac{\theta \lambda {(1-e^{-\theta })}^{-1}}{1-F(x_r;\theta ,\lambda )}{\int }_{x_r}^{\infty }x{e}^{-\lambda x-\theta e^{-\lambda x}}dx,\end{array}$$ at the maximum likelihood estimated values, that is $I_X^{-1}(\widehat{\theta },\widehat{\lambda })$. Subsequently, $100(1-\alpha )\%$ asymptotic confidence intervals for θ and λ are, respectively, given by $\widehat{\theta }\pm Z_{\alpha /2}\sqrt{\text{var}(\widehat{\theta })}$ and $\widehat{\lambda }\pm Z_{\alpha /2}\sqrt{\text{var}(\widehat{\lambda })}$. Here, $Z_{\alpha /2}$ is the upper $(\alpha /2)-$th percentile of the standard normal distribution.

3. Bayesian Estimation

The objective of this section is to obtain Bayes estimators of $(\theta ,\lambda )$ under various loss functions. First, we need to consider a prior for θ and λ to make the desired inference. In the existing literature, Louzada-Neto et al. (2011) considered gamma prior for θ and Jeffery’s prior for λ based on a complete sample observed from PE distribution. The same priors have also been considered by Singh et al. (2014). However in this work, we consider independent gamma priors for both the parameters of the PE distribution, given by, $$\pi (\theta |a_1,b_1) \propto {\theta }^{a_1-1}{e}^{-b_1\theta }, \text{and} \pi (\lambda |a_2,b_2) \propto {\lambda }^{a_2-1}{e}^{-b_2\lambda },$$ where $\pi (\theta |a_1,b_1)$ and $\pi (\lambda |a_2,b_2)$ , respectively, represent the prior distributions for θ and λ. Further, hyper-parameters $a_1,b_1,a_2$, and b₂ are all positive and are responsible for the prior knowledge of $(\theta ,\lambda )$. Therefore, the prior distribution of $(\theta ,\lambda )$ becomes $\pi (\theta ,\lambda )=\pi (\theta |a_1,b_1)\pi (\lambda |a_2,b_2)$. We mention that the prior proposed by Louzada-Neto et al. (2011) can also be seen as a particular case of our proposed prior when the hyper parameter values of a₁ and b₁ are considered zero. Furthermore, non-informative prior $\pi (\theta ,\lambda )=1/(\theta \lambda )$ can also be seen as a particular case corresponds to all the hyper-parameter values that are zero. Now, the posterior distribution of $(\theta ,\lambda )$ given the observed data $\mathit{x}=(x_1,x_2,\dots ,x_r)$ under the proposed prior turn out to have the following form. (9) $$\pi (\theta ,\lambda |\mathrm{x})\propto G_{\lambda }(r+a_1,b_1+\sum _{i=1}^r{x}_i\left)G_{\theta |\lambda }\right(r+a_2,b_2+\sum _{i=1}^r{e}^{-\lambda x_i}\left)Q\right(\theta ,\lambda ),$$(9) where $G_{.}(.,.)$ represents the PDF of gamma distribution, and $Q(\theta ,\lambda )$ is given by, (10) $$Q(\theta ,\lambda )={(1-e^{-\theta })}^{-n}{(1-e^{-\theta e^{-\lambda x_r}})}^{n-r}.$$(10)

Next, we consider squared error loss (SEL) given by ${\delta }_{SEL}(g(\varphi ),\widehat{g}(\varphi ))={(g(\varphi )-\widehat{g}(\varphi ))}^2$, Linex loss (LL), given by ${\delta }_{LL}(g(\varphi ),\widehat{g}(\varphi ))=\hspace{0.17em}\text{exp}(h(\widehat{g}(\varphi )-g(\varphi )))-h(\widehat{g}(\varphi )-g(\varphi ))-1,h\ne 0$, and general entropy loss (EL) function given by ${\delta }_{EL}(g(\varphi ),\widehat{g}(\varphi ))={(\widehat{g}(\varphi )/g(\varphi ))}^q-q\hspace{0.17em}\text{ln}(\widehat{g}(\varphi )/g(\varphi ))-1,q\ne 0$. Note, that the SEL is a symmetric loss function but it may not be appropriate for the situations in which under/over estimation is more serious than the over/under estimation as it puts equal weight to both under as well as over estimations. However, many practical situations require asymmetric loss function for which LL and EL functions can be used. Notice that magnitude of h in LL reflect the degree of asymmetry, such as h < 0, represents the under estimation more serious than the over estimation, and vice versa for h > 0. In a similar way, $q<0(/>0)$ has a more serious effect than positive (/negative) error in EL function. Now, Bayes estimators of a function, say $g(\varphi )$, using the considered prior $\pi (\theta ,\lambda )$ under LL and EL functions are, respectively, given by, (11) $${\widehat{g}}_{LL}(\varphi )=\frac{1}{h}\text{ln}\left(E\right(e^{-hg(\varphi )}\left|\mathit{x}\right)), \text{and} {\widehat{g}}_{EL}(\varphi ) = {\left(E\right({\left(g\right(\varphi \left)\right)}^{-q}\left|\mathit{x}\right))}^{-1/q}.$$(11)

Further, under EL function corresponding to q = −1, it represents the Bayes estimator of $g(\varphi )$ under SEL function. It is seen that the above expressions do not admit closed forms. Therefore, we next use Lindley’s approximation.

3.1. Lindley Approximation

In their work, Lindley (1980) suggested the approximation method to compute Bayes estimates. This method has been used by several authors when the desired Bayes estimators do not admit a closed form. We have briefed the method in the Appendix. Suppose we want to compute Bayes estimates of θ under LL function, then accordingly, we need to compute $E(e^{-h\theta }|\mathrm{x})$. Now following the appendix, we have $g(\varphi )=\theta$ and Bayes estimator turn out to have the following form. $$\begin{array}{c}{\widehat{\theta }}_{LL}=\frac{1}{h}\text{ln}[e^{-h\widehat{\theta }}-h{e}^{-h\widehat{\theta }}({\widehat{\rho }}_1{\widehat{\sigma }}_{12}+{\widehat{\rho }}_2{\widehat{\sigma }}_{22})+0.5[h^2{e}^{-h\widehat{\theta }}{\widehat{\sigma }}_{22}-h{e}^{-h\widehat{\theta }}({\widehat{\sigma }}_{11}{\widehat{\sigma }}_{12}{l}_{30}\\ +{\widehat{\sigma }}_{22}^2{l}_{03}+3{\widehat{\sigma }}_{21}{\widehat{\sigma }}_{22}{l}_{12})]].\end{array}$$

In a similar way, Bayes estimator of θ under EL function is given by $$\begin{array}{c}{\widehat{\theta }}_{EL}=[{\widehat{\theta }}^{-q}-q{\theta }^{-(q+1)}({\widehat{\rho }}_1{\widehat{\sigma }}_{12}+{\widehat{\rho }}_2{\widehat{\sigma }}_{22})+0.5[q(q+1){\theta }^{-(q+1)}{\sigma }_{22}\\ -q{\theta }^{-(q+1)}({\sigma }_{22}^2{l}_{03}+3{\sigma }_{12}{\sigma }_{22}{l}_{12}+{\sigma }_{11}{\sigma }_{21}{l}_{30})]{]}^{-1/q}.\end{array}$$

Notice that all the involved expressions on the right hand side of the above expressions are computed at the maximum likelihood estimates of $(\theta ,\lambda )$, and are reported in the appendix. In a similar way, Bayes estimates of λ can also be obtained, details of all the involved expressions are not reported here for the sake of conciseness. Notice that the method of Lindley can only provide the Bayes estimates but not the associated highest posterior density (HPD) interval estimates. Therefore next, we use importance sampling technique which will be helpful to construct HPD interval estimates using the method of Chen and Shao (1999).

3.2. Importance Sampling

Importance sampling is a very useful technique to draw samples from the associated posterior distribution. In our case, the posterior distribution is given by (9)$$E_3(\theta ,\lambda )=E\left(Z_i{e}^{-\lambda Z_i}\right|Z_i>X_r) = \frac{\lambda \theta {(1-e^{-\theta })}^{-1}}{1-F(x_r;\theta ,\lambda )}{\int }_{x_r}^{\infty }x{e}^{-2\lambda x-\theta e^{-\lambda x}}dx,$$. Observe that a value of λ can be generated from the $G_{\lambda }(r+a_2,b_2+{\sum }_{i=1}^r{x}_i)$ distribution, and further given the generated value of λ the value of θ can be generated from $G_{\theta |\lambda }(r+a_1,b_1+{\sum }_{i=1}^r{e}^{-\lambda x_i})$ distribution. This procedure can be repeated to obtain a sample of $\{({\theta }_1,{\lambda }_1),({\theta }_2,{\lambda }_2),\dots ,({\theta }_s,{\lambda }_s)\}$. Subsequently, Bayes estimates of θ under LL and EL functions are, respectively, given by, $${\widehat{\theta }}_{LL}=\frac{1}{h}\text{ln}\left(\frac{{\sum }_{i=1}^s{e}^{-h{\theta }_i}Q({\theta }_i,{\lambda }_i)}{{\sum }_{i=1}^{s}Q({\theta }_i,{\lambda }_i)}\right), \text{and} {\widehat{\theta }}_{EL} = {\left(\frac{{\sum }_{i=1}^s{\theta }_i^{-q}Q({\theta }_i,{\lambda }_i)}{{\sum }_{i=1}^{s}Q({\theta }_i,{\lambda }_i)}\right)}^{-1/q}.$$

In a similar way, Bayes estimates of λ can also be obtained under the respective loss functions. Now, the generated samples can further help to construct HPD interval estimates for θ and λ using the idea of Chen and Shao (1999). The procedure is also briefly explained in appendix 2 of Singh et al. (2015).

4. Shrinkage Preliminary Test Estimator

In the previous section, we incorporated the prior information using the Bayesian approach. It is to be noticed that the known prior information on some or all of the parameters usually incorporated in the model as a constraint leads to restricted models. However, the validity of a restricted estimator (estimator resulting from restricted model) may be under suspicion and this further leads to need for a preliminary test on the restrictions (see Saleh, 2006; Belaghi et al. 2015a,b for more details and relevant literature cited therein). The objective of this section is to obtain shrinkage and preliminary test estimators. Suppose that there exist some non-sample, information with form of $\theta ={\theta }_0$, and our interest is to estimate θ based on such information. To check the accuracy of this information, one may consider the hypothesis $H_0:\theta ={\theta }_0$ and alternate $H_1:\theta \ne {\theta }_0$. However, it is seen that the construction of shrinkage estimators for θ based on fixed alternatives $H_1:\theta ={\theta }_0+\delta$, for a fixed δ, does not offer substantial performance change compared to $\widehat{\theta }$. In other words, the asymptotic distribution of shrinkage estimator coincides with that of $\widehat{\theta }$ (see Saleh, 2006). Therefore to overcome this situation, we consider local alternatives with form $A_{(m)}:{\theta }_{(r)}={\theta }_0+r^{-1/2}\delta$, where δ is a fixed number, and define the Shrinkage Preliminary Test (SPT) estimator of θ based on maximum likelihood estimates and Bayes estimates, respectively, as, $${\widehat{\theta }}_{M.SPT}=w{\theta }_0+(1-w){\widehat{\theta }}_{MLE}I(W_r<{\chi }_1^2(\gamma \left)\right),$$ and $${\widehat{\theta }}_{B.SPT}=w{\theta }_0+(1-w){\widehat{\theta }}_{\mathit{\text{Bayes}}}I(W_r<{\chi }_1^2(\gamma \left)\right).$$ Here, $w\in [0,1]$. We mention that under $H_0,\sqrt{r(\theta -{\theta }_0)}$ is asymptotically $N(0, \text{Var} (\widehat{\theta }))$, and so the test statistics can be defined as $W_r={(\sqrt{r}(\widehat{\theta }-\theta )/Var(\widehat{\theta }))}^2$. Now, the asymptotic distribution of W_r converges to a non-central chi-square distribution with one degree of freedom and non-centrality parameter ${\Delta }^2/2$, where ${\Delta }^2={\delta }^2/({\delta }^2(\widehat{\theta }))$. Therefore, we reject H₀ when $W_r>{\chi }_1^2(\gamma )$, where γ is type-I error and ${\chi }_1^2(\gamma )$ is the γ-upper quantile of Chi-square distribution with 1 degree of freedom. One may also refer to Belaghi and Asl (2016) for more details. The SPT estimators for λ based on maximum likelihood estimates and Bayes estimates can also be defined in a similar fashion.

5. Prediction

The objective of this section is to predict the lifetimes of the observations $z_i=x_{r+i},i=1,2,\dots ,(n-r)$ censored at the time of r-th failure. Notice, that the conditional density for the $z_i=x_{r+i}>x_r,i=1,2,\dots ,n-r$ can be written as, $$f_1\left(z\right|\mathit{x},\theta ,\lambda )=K\frac{{\left(F\right(z)-F(x_r\left)\right)}^{k-r-1}{(1-F(z\left)\right)}^{n-k}f\left(z\right)}{{(1-F(x_r\left)\right)}^{n-r}},$$ where $K=(k-r)(\begin{array}{c}n-r\\ k-r\end{array}), f(.;\theta )=f(.)$, and $F(.;\theta )=F(.)$. Now, using the binomial expansion the conditional density can be written as (see Singh & Tripathi, 2016), (12) $$f_1\left(z\right|\mathit{x},\theta ,\lambda )=K\sum _{j=0}^{k-r-1}\left(\begin{array}{c}k-r-1\\ j\end{array}\right){(-1)}^{k-r-1-j}{\left[\frac{1-e^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}\right]}^{j-n+r}{\left[\frac{1-e^{-\theta e^{-\lambda z}}}{1-e^{-\theta }}\right]}^{n-r-1-j}f(z),$$(12)

Next, we use different predictors to predict the censored observations.

5.1. Best Unbiased Predictor

This section deals with the best unbiased predictor (BUP) to predict z observations. Notice, that a statistic $\widehat{z}$ used to predict z is called a BUP of z, if the predictor error $(\widehat{z}-z)$ has a mean zero, and its prediction error variance var$(\widehat{z}-z)$ is less than or equal to that of any other unbiased predictor of z. Therefore, the BUP of z is $\widehat{z}=E[z|X_r=x_r]$. Subsequently, using the conditional density given by (12)(12) $$f_1\left(z\right|\mathit{x},\theta ,\lambda )=K\sum _{j=0}^{k-r-1}\left(\begin{array}{c}k-r-1\\ j\end{array}\right){(-1)}^{k-r-1-j}{\left[\frac{1-e^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}\right]}^{j-n+r}{\left[\frac{1-e^{-\theta e^{-\lambda z}}}{1-e^{-\theta }}\right]}^{n-r-1-j}f(z),$$(12) , the BUP of z is given by, $$\begin{array}{c}\widehat{z}={\int }_{x_r}^{\infty }z{f}_1\left(z\right|\mathit{x},\theta ,\lambda )dz, z=x_{r+i}>x_r, i=1,2,\dots ,n-r.\\ =-\frac{K}{\lambda }\sum _{j=0}^{k-r-1}\left(\begin{array}{c}k-r-1\\ j\end{array}\right){(-1)}^{k-r-1-j}{\left[\frac{1-e^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}\right]}^{j-n+r}\\ \times {\int }_{F\left(x_r\right)}^1\hspace{0.17em}\text{ln}\hspace{0.17em}[-\frac{1}{\theta }\text{ln}((1-v)(1-e^{-\theta })-1\left)\right]{(1-v)}^{n-r-1-j}dv.\end{array}$$

Notice, that we put $u=[1-e^{-\theta e^{-\lambda z}}]/[1-e^{-\theta e^{-\lambda x_r}}]$ in the condition density given by (12)(12) $$f_1\left(z\right|\mathit{x},\theta ,\lambda )=K\sum _{j=0}^{k-r-1}\left(\begin{array}{c}k-r-1\\ j\end{array}\right){(-1)}^{k-r-1-j}{\left[\frac{1-e^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}\right]}^{j-n+r}{\left[\frac{1-e^{-\theta e^{-\lambda z}}}{1-e^{-\theta }}\right]}^{n-r-1-j}f(z),$$(12) to obtain the above expression, and we observe that the probability density of $u|\mathit{x}\sim \text{Beta} (n-r-k+1,k)$ distribution. Finally, the involved θ and λ in the above expression can be replaced by their respective maximum likelihood estimates to obtain BUP of $z=x_{r+i},i=1,2,\dots ,n-r$.

5.2. Conditional Median Predictor

This section deals with obtaining the conditional median predictor (CMP) proposed by Raqab and Nagaraja (1995). A predictor $\widehat{z}$ is called CMP of z if it is the median of the conditional density of z given the x_r observation, that is $P(z\le \widehat{z}|x_r)=P(z\ge \widehat{z}|x_r)$. Now, observe that the random variable $w=(1-u)|\mathit{x}\sim \text{Beta}(k,n-r-k+1)$ distribution, and subsequently, using the distribution of w, CMP of $z(=x_{r+i},i=1,2,\dots ,n-r)$, is given by, $$\widehat{z}=-\frac{1}{\lambda }\text{ln}[-\frac{1}{\theta }\text{ln}(M(1-e^{-\theta })-1\left)\right],$$ where M stands for median of the Beta$(k,n-r-k+1)$ distribution. Further, the associated 100$(1-\alpha )$% prediction interval can be obtained as, $$(-\frac{1}{\lambda }\text{ln}[-\frac{1}{\theta }\text{ln}\left(B_{\alpha /2}\right(1-e^{-\theta })-1)],-\frac{1}{\lambda }\text{ln}[-\frac{1}{\theta }\text{ln}\left(B_{1-\alpha /2}\right(1-e^{-\theta })-1)\left]\right),$$ where B_p represents the 100-pth percentile of the Beta$(k,n-r-k+1)$ distribution. Further, observe that the PDF of random variable w is unimodal function of w for $1<k<(n-r)$. Subsequently, the $100(1-\alpha )\%$ highest conditional density (HCD) prediction interval is, $$(-\frac{1}{\lambda }\text{ln}[-\frac{1}{\theta }\text{ln}\left(w_1\right(1-e^{-\theta })-1)],-\frac{1}{\lambda }\text{ln}[-\frac{1}{\theta }\text{ln}\left(w_2\right(1-e^{-\theta })-1)\left]\right).$$

Here, w₁ and w₂ are the solution to the following equations. $${\int }_{w_1}^{w_2}g\left(w\right)dw=1-\alpha ,\hspace{1em} \text{and} \hspace{1em}g\left(w_1\right)=g\left(w_2\right).$$

More precisely, the above equations can be rewritten as, $${\text{Beta}}_{w_2}(n-r-k+1,k)-{ \text{Beta}}_{w_1}(n-r-k+1,k)=1-\alpha ,\hspace{0.5em}\text{and}\hspace{0.5em}{\left(\frac{1-w_1}{1-w_2}\right)}^{n-r-k}={\left(\frac{w_2}{w_1}\right)}^{k-1}$$

Notice, that $\text{Beta}(.,.)$ represents the CDF of $\text{Beta}(.,.)$ distribution.

5.3. Bayesian Prediction

This section deals with predictive estimates and associated predictive interval estimates using Bayesian approach. Notice, that under the considered prior $\pi (\theta ,\lambda )$ the posterior predictive density is given by, $$f_1^{*}(\theta ,\lambda |\mathrm{x})={\int }_0^{\infty }{\int }_0^{\infty }{f}_1(z|\mathit{x},\theta ,\lambda )\pi (\theta ,\lambda |\mathit{x})d\theta d\lambda ,$$

Now, the Bayesian predictive estimate of z under LL and EL are given by, (13) $${\widehat{z}}_{LL}=\frac{1}{h}\text{ln}\left[{\int }_{x_r}^{\infty }{e}^{-hz}{f}_1^{*}\right(\theta ,\lambda \left|\mathit{x}\right)dz] = \frac{1}{h}\text{ln}[{\int }_0^{\infty }{\int }_0^{\infty }{I}_1(\theta ,\lambda )\pi (\theta ,\lambda |\mathit{x})d\theta d\lambda ],$$(13) (14) $${\widehat{z}}_{EL}={\left[{\int }_{x_r}^{\infty }{z}^{-q}{f}_1^{*}\right(\theta ,\lambda \left|\mathit{x}\right)dz]}^{-1/q} = {\left[{\int }_0^{\infty }{\int }_0^{\infty }{I}_2\right(\theta ,\lambda )\pi (\theta ,\lambda \left|\mathit{x}\right)d\theta d\lambda ]}^{-1/q},$$(14) where, $$\begin{array}{c}{I}_1(\theta ,\lambda )={\int }_{x_r}^{\infty }{e}^{-hz}{f}_1\left(z\right|\mathit{x},\theta ,\lambda )dz\\ =-\frac{K}{\lambda }\sum _{j=0}^{k-r-1}\left(\begin{array}{c}k-r-1\\ j\end{array}\right){(-1)}^{k-r-1-j}{\left[\frac{1-e^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}\right]}^{j-n+r}\\ \times {\int }_{F\left(x_r\right)}^1\hspace{0.17em}\text{exp}\hspace{0.17em}\{-h[\text{ln}[-\frac{1}{\theta }\text{ln}((1-v)(1-e^{-\theta })-1\left)\right]\left]\right\}{(1-v)}^{n-r-1-j}dv,\end{array}$$ and $$\begin{array}{c}{I}_2(\theta ,\lambda )={\int }_{x_r}^{\infty }{z}^{-q}{f}_1\left(z\right|\mathit{x},\theta ,\lambda )dz\\ =-\frac{K}{\lambda }\sum _{j=0}^{k-r-1}\left(\begin{array}{c}k-r-1\\ j\end{array}\right){(-1)}^{k-r-1-j}{\left[\frac{1-e^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}\right]}^{j-n+r}\\ \times {\int }_{F\left(x_r\right)}^1\{\hspace{0.17em}\text{ln}[-\frac{1}{\theta }\text{ln}((1-v)(1-e^{-\theta })-1\left)\right]{\}}^{-q}{(1-v)}^{n-r-1-j}dv.\end{array}$$

Now observe that the expression given by (13)(13) $${\widehat{z}}_{LL}=\frac{1}{h}\text{ln}\left[{\int }_{x_r}^{\infty }{e}^{-hz}{f}_1^{*}\right(\theta ,\lambda \left|\mathit{x}\right)dz] = \frac{1}{h}\text{ln}[{\int }_0^{\infty }{\int }_0^{\infty }{I}_1(\theta ,\lambda )\pi (\theta ,\lambda |\mathit{x})d\theta d\lambda ],$$(13) and (14)(14) $${\widehat{z}}_{EL}={\left[{\int }_{x_r}^{\infty }{z}^{-q}{f}_1^{*}\right(\theta ,\lambda \left|\mathit{x}\right)dz]}^{-1/q} = {\left[{\int }_0^{\infty }{\int }_0^{\infty }{I}_2\right(\theta ,\lambda )\pi (\theta ,\lambda \left|\mathit{x}\right)d\theta d\lambda ]}^{-1/q},$$(14) can be, respectively, seen as $\frac{1}{h}\text{ln}[E(I_1(\theta ,\lambda )|\mathit{x})]/h$ and ${[E(I_2(\theta ,\lambda )|\mathit{x})]}^{-1/q}$. Subsequently, samples drawn from the posterior distribution given by (9)(9) $$\pi (\theta ,\lambda |\mathrm{x})\propto G_{\lambda }(r+a_1,b_1+\sum _{i=1}^r{x}_i\left)G_{\theta |\lambda }\right(r+a_2,b_2+\sum _{i=1}^r{e}^{-\lambda x_i}\left)Q\right(\theta ,\lambda ),$$(9) using importance sampling procedure as mentioned in Section 3.2 can be used to compute the expressions. Further, predictive survival function is given by, $$S^{*}\left(t\right|\mathit{x})={\int }_0^{\infty }{\int }_0^{\infty }{S}_1(t|\mathit{x},\theta ,\lambda )\pi (\theta ,\lambda |\mathit{x}),$$ where, $$S_1\left(t\right|\mathit{x},\theta ,\lambda )=\frac{P(X>t|\mathit{x},\theta ,\lambda )}{P(X>x_r|\mathit{x},\theta ,\lambda )} = {\left[\frac{1-e^{-\theta e^{-\lambda t}}}{1-e^{-\theta e^{-\lambda x_r}}}\right]}^{n-r}$$ Now, the 100$(1-\alpha )$% equal-tail credible predictive interval estimates can be obtained on solving the following expressions for the lower bound L and upper bound U, $$S_1^{*}\left(L\right|\mathit{x})=1-\frac{\alpha }{2}, \text{and} S_1^{*}(U\left|\mathit{x}\right) = \frac{\alpha }{2}.$$

One may further refer to the algorithm given by Singh and Tripathi (2016) to solve the above expressions.

6. Data Analysis

In this section, we analyze a real data set taken from Pepi (1994). This data set represents the failure lifetimes of all-glass airplane window design that measure polished window strength. The failure lifetimes are given below.

18.83, 20.80, 21.657, 23.03, 23.23, 24.05, 24.321, 25.5, 25.52, 25.8, 26.69,

26.77, 26.78, 27.05, 27.67, 29.90, 31.11, 33.2, 33.73, 33.76, 33.89, 34.76,

35.75, 35.91, 36.98, 37.08, 37.09, 39.58, 44.045, 45.29, and 45.381.

In their work, Singh et al. (2014) considered this data set, and used a graphical method to test the goodness-of-fit to the PE distribution. We consider the fitting of this data to the PE distribution using “fitdistrplus” package in R-statistical language. We further compare the fitness of this data set with other lifetime distributions, such as Weibull distribution (WD), generalized exponential (GE), and Burr XII distributions based on Akaike’s information criterion (AIC), Bayesian information criterion (BIC), and Kolmogrov-Smirnov (KS) test statistic value. We mention that the better model will correspond to the minimum value of AIC, BIC, and KS test statistics, and Table 1 suggests that the PE distribution fits this data set better as compared to the other considered distributions. Next, from the given data set we generate two different type-II censored samples correspond to r = 23 and r = 28, and based on these samples we first estimate the unknown parameters of the PE distribution using classical and Bayesian approaches. Table 2 reports maximum likelihood estimates computed using NR and EM algorithms, and non-informative Bayes estimates obtained using Lindley and importance sampling (Is) against LINEX, SEL, and GEL functions. It is observed that estimates obtained using the NR method are generally larger compared to estimates using the EM algorithm, and Bayes estimated values of θ using importance sampling (Is) are smaller as compared to Lindley method but for λ both are close to each other. Further, Table 3 reports the associated 95% asymptotic confidence and HPD interval estimates, and it is seen that the HPD interval estimates have smaller intervals as compared to asymptotic confidence interval estimates. In Table 4, we report shrinkage preliminary test estimates based on maximum likelihood and Bayes estimates, and in the process, we considered ${\lambda }_0=0.022$ and ${\theta }_0=0.5$ as non-sample prior information with type-I error 0.05 and w = 0.5. Further, Table 5 reports first predictive observations obtained using different proposed methods, and Table 6 reports predictive interval estimates. Tabulated values suggest that the predictive estimates obtained using classical predictors are closer to the true values than the other predictors. Further, the reported predictive interval estimates contain the true observations, and it can be observed that the interval estimates using the equal-tail method have smaller intervals as compared to the pivotal and HCD methods.

Table 1. Goodness-of-fit tests for given distributions.

Model	PDF	$\widehat{\theta }$	$\widehat{\lambda }$	NLC	AIC	BIC	K-S
PE	$f(x;\theta ,\lambda )$	96.955	0.1669	104.1426	212.2852	215.1531	0.1041
WD	$\frac{\theta }{{\lambda }^{\theta }}{x}^{\theta -1}{e}^{{(-\frac{x}{\lambda })}^{\theta }}$	4.6349	33.673	105.4889	214.9778	217.8458	0.9677
GE	$\theta \lambda e^{-\lambda x}{(1-e^{-\lambda x})}^{\theta }$	93.849	0.1660	104.1543	212.2986	215.1666	0.1047
Burr XII	$\theta \lambda x^{\lambda -1}{(1+x^{\lambda })}^{-\theta -1}$	4.2567	0.0690	174.3869	352.7738	355.6418	0.8890

Table 2. ML and Bayes estimates for the real data set.

r	Parameter	Maximum likelihood estimates		Bayes estimates
				Method	LINEX		SEL	GEL
		NR	EM		$h=$ −0.25	$h=$ 0.5		$q=$ −0.5	$q=$ 1.25
23	λ	0.1360	0.0247	Lindley	0.0197	0.0199	0.0211	0.0205	0.0198
	θ	53.689	0.5683		0.9566	0.9620	0.9908	0.9777	0.9631
	λ			Is	0.0185	0.0182	0.0184	0.0183	0.0177
	θ				0.8696	0.8624	0.8672	0.8621	0.8461
28	λ	0.1474	0.0287	Lindley	0.0311	0.0296	0.0343	0.0316	0.0299
	θ	64.644	0.6000		1.7888	2.0553	2.0030	1.9562	2.0012
	λ			Is	0.0201	0.0200	0.0200	0.0199	0.0197
	θ				1.0261	1.0111	1.0206	1.0123	0.9895

Table 3. 95% asymptotic confidence and HPD interval estimates for the real data set.

r	Parameter	Asymptotic confidence interval	HPD interval
23	λ	(0.0239–0.0500)	(0.0105–0.0245)
	θ	(0.5671–0.6820)	(0.6578–1.1387)
28	λ	(0.0278–0.0583)	(0.0103–0.0258)
	θ	(0.5889–0.7201)	(0.5032–1.4201)

Table 4. Shrinkage preliminary test estimates for the real data set.

r	Parameter	MLE	LINEX		SEL	GEL
			$h=-0.25$	h = 0.5		$q=-0.5$	q = 1.25
23	λ	0.0223	0.0202	0.0201	0.0202	0.0201	0.0199
	θ	0.5500	0.6848	0.6812	0.6836	0.6810	0.6731
28	λ	0.0254	0.0210	0.0209	0.0210	0.0209	0.0208
	θ	0.5750	0.7631	0.7556	0.7603	0.7561	0.7448

Table 5. Predictive estimates of $x_{r+1}$ for the real data set.

r	True value	Classical		Bayesian prediction
		prediction		LINEX			GEL
		BUP	CMP	$h=-0.25$	h = 0.5	SEL	$q=-0.5$	q = 1.25
23	35.91	38.4258	37.8246	36.8184	36.9971	37.6044	37.8117	35.2086
28	44.045	47.3821	46.0993	46.9322	44.7738	46.6577	47.2253	45.1479

Table 6. Predictive interval estimates of $x_{r+1}$ for the real data set.

r	True value	Pivotal	HCD	Equal-tail
23	35.91	(34.9888–37.6267)	(33.5195–36.7500)	(34.9208–37.4998)
28	44.045	(42.8820–46.6814)	(41.3279–45.5800)	(43.3279–46.3809)

7. Simulation Study

In this section, we conduct simulation study for different choices of n such as n = 30, 50, 100 and r such as $r=20,23,25,35,40,50,85,90,95$ correspond to the true parameter values of $(\theta ,\lambda )$ as $(2,0.5)$. We first generate n number of observations from PE distribution, and after sorting the generated lifetimes we consider first r observations to replicate the data under type-II censoring. We mention that R-statistical language has been used for this purpose. In Table 7, we report the maximum likelihood estimated values. It can be seen that average estimated values obtained using the NR method are generally higher as compared to the EM algorithm. Further, the associated MSE values obtained using the NR method are larger following the EM algorithm. It is further observed that the higher values of n and r lead to better estimates in the sense close to the true parameter values and having smaller MSE values. Table 8 presents the average estimates and associated MSE values for the Bayes estimates using both non-informative (NIN) and informative (IN) priors. Notice, that to obtain the hyper-parameter values under IN prior, we first generate 1,000 number of complete samples from PE distribution with λ = 2 and $\theta =0.5$, each sample with 30 observations. Now corresponding to each sample, we first obtain the maximum likelihood estimates of λ and θ, and then compare the mean and variance of these samples with the mean and variance of the considered priors (see Dey et al. (2016); Singh and Tripathi, 2018 for more details). Subsequently, we get the hyper-parameter values as $a_1=16.5443,b_1=7.3979,a_2=1.2145$, and $b_2=1.3608$. The results show that the performance of IN prior is better than the NIN prior both in terms of average estimates and MSE values. The performance of Bayes estimates obtained using importance sampling technique is also good as compared to the estimates obtained using the Lindely method. It is further observed that as expected, the higher value of h and q, respectively, under LINEX and GEL functions lead to smaller estimates as compared to the negative values of the h and q. Further, in Table 9 we present the 95% asymptotic confidence and highest posterior density interval estimates along with average interval lengths (AILs). From tabulated values, it is observed that the highest posterior density interval estimates obtained under IN prior have smaller AILs as compared to NIN priors.

Table 7. Average and MSE values of maximum likelihood estimates.

n	r	λ				θ
		NR		EM		NR		EM
		Avg	MSE	Avg	MSE	Avg	MSE	Avg	MSE
30	20	1.9172	0.2683	1.2380	0.5955	0.5881	0.0785	0.3736	0.0160
	23	1.8054	0.2878	1.3926	0.3908	0.5157	0.0812	0.4254	0.0057
	25	1.7603	0.3110	1.5176	0.2605	0.4978	0.7700	0.4529	0.0023
50	35	1.7863	0.1914	1.5161	0.1240	0.5964	0.0781	0.3774	0.0152
	40	1.6691	0.2572	1.4028	0.3702	0.5086	0.0795	0.4435	0.0034
	45	1.6553	0.2830	1.5535	0.2344	0.4791	0.0768	0.4558	0.0020
100	85	1.7527	0.1028	1.4610	0.2974	0.4785	0.0753	0.4540	0.0021
	90	1.7631	0.1023	1.5365	0.2289	0.4508	0.0725	0.4565	0.0019
	95	1.8172	0.0913	1.7063	0.1060	0.4914	0.0573	0.4550	0.0209

Table 8. Average and MSE values of Bayes estimates.

n	r	Method			NIN Prior					IN Prior
					LINEX			GEL		LINEX			GEL
					$h=-$0.25	$h=$0.5	SEL	$q=$−0.5	$q=$1.25	$h=$−0.25	$h=$0.5	SEL	$q=$−0.5	$q=$1.25
30	20	Lin	λ	Avg	1.6343	1.6123	1.6194	1.6222	1.5737	1.8505	1.8362	1.8505	1.8549	1.8208
				MSE	0.3485	0.4102	0.3568	0.3548	0.4420	0.2405	0.2303	0.2405	0.3490	0.2309
			θ	Avg	0.5752	0.5477	0.5520	0.5707	0.5213	0.5494	0.5423	0.5494	0.5334	0.5119
				MSE	0.0889	0.0828	0.0834	0.0885	0.0897	0.0763	0.0843	0.0763	0.0736	0.0885
		Is	λ	Avg	1.6722	1.6648	1.6738	1.6750	1.6681	1.7377	1.7400	1.7377	1.7232	1.7224
				MSE	0.1176	0.1148	0.1429	0.1206	0.1440	0.0864	0.0853	0.0864	0.0920	0.0676
			θ	Avg	0.5230	0.5058	0.5126	0.5388	0.5034	0.5213	0.4917	0.5213	0.5271	0.4872
				MSE	0.0094	0.0013	0.0095	0.0284	0.0012	0.0055	0.0045	0.0055	0.0047	0.0045
	23	Lin	λ	Avg	1.5668	1.5434	1.5514	1.5714	1.5491	1.9277	1.8443	1.8698	1.7887	1.7880
				MSE	0.4045	0.3678	0.3924	0.3940	0.3910	0.3292	0.3527	0.3489	0.3793	0.3463
			θ	Avg	0.5397	0.5069	0.5109	0.5233	0.5055	0.5079	0.4504	0.5044	0.5180	0.4761
				MSE	0.0841	0.0829	0.0981	0.0820	0.0899	0.0817	0.0812	0.0706	0.0809	0.0811
		Is	λ	Avg	1.7653	1.7458	1.7522	1.7663	1.7385	1.7821	1.7804	1.7899	1.7716	1.7657
				MSE	0.0982	0.0960	0.0967	0.1009	0.0976	0.0721	0.0612	0.0702	0.0753	0.0713
			θ	Avg	0.4800	0.4478	0.4506	0.4766	0.4551	0.4953	0.4941	0.4971	0.4847	0.4840
				MSE	0.0043	0.0022	0.0042	0.0046	0.0023	0.0040	0.0041	0.0040	0.0039	0.0040
	25	Lin	λ	Avg	1.5877	1.5662	1.5778	1.5903	1.5580	1.8391	1.7894	1.7971	1.6830	1.6743
				MSE	0.3731	0.3546	0.3037	0.3357	0.3653	0.3673	0.3255	0.3054	0.3244	0.3428
			θ	Avg	0.5538	0.5178	0.5366	0.5437	0.5209	0.5207	0.4852	0.5122	0.5280	0.4778
				MSE	0.0818	0.0915	0.1090	0.0778	0.0856	0.0725	0.0747	0.1047	0.0769	0.0836
		Is	λ	Avg	1.7793	1.7931	1.7886	1.7610	1.7886	1.7636	1.7757	1.8230	1.8117	1.8109
				MSE	0.0850	0.0835	0.0840	0.0847	0.0847	0.0597	0.0576	0.0583	0.0625	0.0592
			θ	Avg	0.5091	0.4838	0.4813	0.5170	0.5170	0.5200	0.5229	0.5219	0.5095	0.5087
				MSE	0.0047	0.0042	0.0048	0.0044	0.0023	0.0042	0.0039	0.0040	0.0041	0.0046
50	35	Lin	λ	Avg	1.6208	1.6120	1.6180	1.6353	1.6089	1.8916	1.8674	1.8733	1.8823	1.8410
				MSE	0.3009	0.2713	0.3107	0.3186	0.2934	0.1071	0.1069	0.1066	0.2011	0.1070
			θ	Avg	0.5616	0.5438	0.5599	0.5756	0.5537	0.5475	0.5134	0.5445	0.5591	0.5353
				MSE	0.0752	0.0722	0.0819	0.0736	0.0797	0.0667	0.0654	0.4390	0.0720	0.0704
		Is	λ	Avg	1.6442	1.6235	1.6251	1.6309	1.6139	1.6658	1.6704	1.6688	1.6613	1.6671
				MSE	0.1423	0.1404	0.1912	0.1444	0.1920	0.1240	0.1218	0.1225	0.1264	0.1234
			θ	Avg	0.4998	0.4737	0.4803	0.4882	0.4630	0.5145	0.5154	0.5151	0.5109	0.5114
				MSE	0.0053	0.0080	0.0053	0.0055	0.0077	0.0028	0.0028	0.0028	0.0026	0.0027
	40	Lin	λ	Avg	1.6557	1.6021	1.6408	1.6868	1.6251	1.8341	1.8045	1.8237	1.7942	1.7609
				MSE	0.3000	0.2744	0.2775	0.3170	0.2914	0.2596	0.2253	0.2107	0.3038	0.2407
			θ	Avg	0.5886	0.5447	0.5658	0.5760	0.5315	0.5142	0.4879	0.5138	0.5316	0.5109
				MSE	0.0771	0.0795	0.1023	0.0711	0.0787	0.0732	0.0753	0.0703	0.0740	0.0682
		Is	λ	Avg	1.6817	1.6766	1.6799	1.6769	1.6761	1.7311	1.7377	1.7355	1.7248	1.7330
				MSE	0.1178	0.1156	0.1280	0.1201	0.1031	0.0896	0.0871	0.0879	0.0922	0.0889
			θ	Avg	0.5061	0.5044	0.5066	0.5035	0.5034	0.5421	0.4978	0.5427	0.5385	0.4966
				MSE	0.0019	0.0022	0.0020	0.0019	0.0023	0.0013	0.0021	0.0017	0.0020	0.0021
	45	Lin	λ	Avg	1.6512	1.6101	1.6365	1.6433	1.6173	1.8818	1.8459	1.8321	1.6880	1.6630
				MSE	0.3068	0.2832	0.2756	0.3196	0.3133	0.2403	0.2389	0.2439	0.3081	0.2954
			θ	Avg	0.5737	0.5501	0.5683	0.5849	0.5430	0.5253	0.5099	0.5169	0.5200	0.4990
				MSE	0.0767	0.0774	0.0822	0.0804	0.0690	0.0738	0.0635	0.0713	0.0787	0.0666
		Is	λ	Avg	1.7678	1.7267	1.7537	1.7598	1.7305	1.8019	1.8171	1.8080	1.7936	1.8047
				MSE	0.0849	0.0820	0.0835	0.0872	0.0820	0.0641	0.0619	0.0626	0.0665	0.0635
			θ	Avg	0.6219	0.5788	0.6125	0.6186	0.5878	0.5412	0.5203	0.5417	0.5377	0.5191
				MSE	0.0180	0.0026	0.0182	0.0171	0.0025	0.0040	0.0026	0.0041	0.0037	0.0025
100	85	Lin	λ	Avg	1.6028	1.5707	1.6150	1.6362	1.5856	1.7041	1.6533	1.6584	1.6647	1.6343
				MSE	0.2824	0.2771	0.2600	0.2838	0.2820	0.2586	0.3621	0.2704	0.2719	0.2610
			θ	Avg	0.5363	0.5122	0.5154	0.5408	0.5026	0.5285	0.4993	0.5016	0.5164	0.4924
				MSE	0.0812	0.0869	0.0986	0.0739	0.0690	0.0826	0.0813	0.0742	0.0696	0.0654
		IS	λ	Avg	1.6493	1.5818	1.6209	1.6467	1.6100	1.6815	1.6845	1.6835	1.6785	1.6824
				MSE	0.1329	0.1316	0.1320	0.1344	0.1311	0.1144	0.1130	0.1135	0.1160	0.1140
			θ	Avg	0.5790	0.5468	0.5471	0.5463	0.5657	0.5282	0.5395	0.5283	0.5274	0.5394
				MSE	0.0033	0.0033	0.0033	0.0032	0.0032	0.0018	0.0026	0.0018	0.0017	0.0026
	90	Lin	λ	Avg	1.6317	1.5892	1.6241	1.6591	1.5880	1.6814	1.6566	1.6609	1.6744	1.6007
				MSE	0.3002	0.3355	0.2968	0.2991	0.2904	0.2275	0.2997	0.2816	0.2822	0.2940
			θ	Avg	0.5638	0.5139	0.5477	0.5520	0.5021	0.5392	0.5125	0.5382	0.5441	0.5376
				MSE	0.0740	0.0893	0.0766	0.0677	0.0756	0.0734	0.0691	0.0688	0.0604	0.0743
		IS	λ	Avg	1.7210	1.7004	1.7034	1.7174	1.7020	1.7242	1.7280	1.7267	1.7203	1.7253
				MSE	0.1058	0.1043	0.1048	0.1074	0.1041	0.0920	0.0904	0.0910	0.0938	0.0916
			θ	Avg	0.5204	0.4800	0.4803	0.4997	0.4796	0.4895	0.4896	0.4890	0.4887	0.4894
				MSE	0.0011	0.0010	0.0011	0.0012	0.0012	0.0009	0.0009	0.0009	0.0001	0.0009
	95	Lin	λ	Avg	1.5882	1.5011	1.5688	1.5710	1.5596	2.0539	1.9187	1.9780	1.9868	1.9165
				MSE	0.5155	0.5133	0.4868	0.5629	0.5095	0.5034	0.1606	0.1956	0.5400	0.2233
			θ	Avg	0.5337	0.0773	0.5147	0.5435	0.4912	0.5097	0.4893	0.4958	0.5193	0.4853
				MSE	0.0792	0.0773	0.0670	0.0864	0.0812	0.0701	0.0789	0.0646	0.0823	0.0801
		IS	λ	Avg	1.7729	1.7513	1.7658	1.7886	1.7442	1.7704	1.7751	1.7735	1.7658	1.7718
				MSE	0.0820	0.0804	0.0810	0.0837	0.0816	0.0716	0.0699	0.0705	0.0734	0.0711
			θ	Avg	0.5234	0.5035	0.5135	0.5327	0.5123	0.5308	0.5307	0.5310	0.5301	0.5308
				MSE	0.0015	0.0021	0.0023	0.0024	0.0020	0.0010	0.0013	0.0020	0.0019	0.0018

Table 9. $95\%$ asymptotic confidence and HPD interval estimates.

n	r		Asymptotic confidence	HPD interval
			interval/AIL	NIN prior/AIL	IN prior/AIL
30	20	λ	(0.8487–2.0884)/1.2397	(0.5392–2.0477)/1.5085	(1.5373–2.1924)/0.6551
		θ	(0.2666–0.5461)/0.2795	(0.2257–0.5170)/0.2913	(0.3240–0.5128)/0.1888
	23	λ	(0.9902–2.0841)/1.0939	(0.5895–2.1046)/1.5151	(1.5378–2.2958)/0.7580
		θ	(0.2875–0.5635)/0.2760	(0.2398–0.6552)/0.4154	(0.3323–0.5061)/0.1739
	25	λ	(1.0981–2.3215)/1.2234	(0.6289–2.0114)/1.3825	(1.6377–2.3964)/0.7587
		θ	(0.2841–0.6219)/0.3378	(0.2103–0.7499)/0.5396	(0.3351–0.5189)/0.1838
50	35	λ	(0.9257–2.1151)/1.1894	(0.3732–2.0080)/1.6348	(1.1253–2.0722)/0.9469
		θ	(0.2930–0.5370)/0.2440	(0.2213–0.6089)/0.3876	(0.3367–0.5289)/0.1922
	40	λ	(1.0902–2.1174)/1.0272	(0.3804–2.0656)/1.6852	(1.0879–2.1875)/1.0996
		θ	(0.3288–0.5583)/0.2295	(0.3084–0.7501)/0.4417	(0.3147–0.5660)/0.2513
	45	λ	(1.2835–2.0576)/0.7741	(0.4571–2.2195)/1.7624	(1.1002–2.3603)/1.2601
		θ	(0.2960–0.6158)/0.3198	(0.3256–0.7191)/0.3935	(0.3173–0.5941)/0.2768
100	85	λ	(1.2235–2.1999)/0.9764	(0.2116–2.0584)/1.8478	(1.3116–2.0795)/0.7679
		θ	(0.3624–0.5456)/0.1832	(0.1330–0.8380)/0.7050	(0.3279–0.5445)/0.2266
	90	λ	(1.3359–2.1107)/0.7748	(0.9867–2.1474)/1.1607	(1.4276–2.0795)/0.6519
		θ	(0.3470–0.5660)/0.2190	(0.2336–0.6599)/0.4263	(0.3269–0.5005)/0.1746
	95	λ	(1.4959–2.2097)/0.7138	(1.2895–2.2869)/1.0034	(1.6088–2.1009)/0.4021
		θ	(0.3177–0.5924)/0.2747	(0.1240–0.6664)/0.5424	(0.3377–0.5427)/0.2050

Next, to obtain shrinkage preliminary test estimates, we first calculate the test statistic W_r under the null hypothesis, $H_0:\theta ={\theta }_0$. In the process, we considered w = 0.5 and the type-I error equal to 0.05, and assume ${\lambda }_0=2.2$ and ${\theta }_0=0.45$ for some non-sample prior information. We then make use of maximum likelihood estimates obtained using the EM algorithm for maximum likelihood based shrinkage preliminary test estimates, and Bayes estimates obtained using importance sampling for Bayes estimates based on shrinkage preliminary test estimates. All the estimates are reported in Table 10, and to compare these estimates we use relative efficiency (RE). Note, that the relative efficiency (RE) of the suggested estimator, say $\tilde{\theta }$ to the estimator $\widehat{\theta }$ is defined by $RE(\tilde{\theta }:\widehat{\theta })=MSE(\widehat{\theta })/MSE(\tilde{\theta })$, and RE larger than one indicates the degree of superiority of the estimator $\tilde{\theta }$ over $\widehat{\theta }$. It is observed that Bayes shrinkage preliminary test estimates have higher relative efficiencies than the maximum likelihood based estimates. Further, the performance of Bayes shrinkage preliminary test estimates of λ under LINEX loss is found satisfactory than the other Bayes shrinkage preliminary test estimates. Table 11 reports the first classical and Bayesian predictive estimated values. The Bayesian predictive estimates are computed under both NIN and IN priors. Tabulated values suggest that the predictive observations using CMP are generally smaller than BUP and Bayes predictors. Finally, in Table 12 we report the predictive interval estimates along with AILs. It is observed that predictive intervals using the HCD method have smaller AILs as compared to the pivotal method, and predictive HPD intervals have smaller AILs under IN prior as compared to NIN prior.

Table 10. Average and relative efficiency values of shrinkage preliminary test estimates.

n	r				NIN Prior					IN Prior
				MLE	LINEX			GEL		LINEX			GEL
				MLE	$h=$−0.25	$h=$0.5	SEL	$q=$−0.5	$q=$1.25	$h=$−0.25	$h=$0.5	SEL	$q=$−0.5	$q=$1.25
30	20	λ	Avg	1.8919	1.7450	1.7393	1.7308	1.7288	1.7340	1.7934	1.7855	1.7907	1.7879	1.7783
			RE	1.0077	1.2481	1.2509	1.1909	1.2509	1.2544	1.2248	1.2320	1.2273	1.2270	1.2357
		θ	Avg	0.5257	0.4907	0.4889	0.4901	0.4879	0.4817	0.4879	0.5113	0.4870	0.4840	0.5010
			RE	1.3795	1.1609	1.1571	1.1596	1.1746	1.1600	1.1804	1.1906	1.1793	1.2183	1.1900
	23	λ	Avg	1.7582	1.7828	1.7738	1.7798	1.7766	1.7660	1.8250	1.8146	1.8214	1.8178	1.8054
			RE	1.0047	1.1381	1.1445	1.1403	1.1400	1.1476	1.1171	1.1227	1.1190	1.1164	1.1236
		θ	Avg	0.4945	0.4969	0.4952	0.4964	0.4944	0.4887	0.4589	0.4885	0.4582	0.4558	0.4797
			RE	1.4193	1.2319	1.2301	1.2346	1.2149	1.1945	1.1434	1.1572	1.1429	1.1820	1.1528
	25	λ	Avg	1.6962	1.8201	1.8084	1.8161	1.8121	1.7986	1.8611	1.8485	1.8568	1.8525	1.8379
			RE	1.0025	1.1161	1.1196	1.1174	1.1139	1.1190	1.0966	1.0992	1.0976	1.0970	1.1010
		θ	Avg	0.4809	0.5008	0.4991	0.5002	0.4982	0.4924	0.5111	0.5448	0.5103	0.5076	0.5354
			RE	1.4301	1.2292	1.2322	1.2396	1.2281	1.2038	1.2783	1.3587	1.2764	1.2506	1.3405
50	35	λ	Avg	1.7820	1.6714	1.6685	1.6487	1.6476	1.6656	1.7041	1.7000	1.7028	1.7012	1.6959
			RE	1.0016	1.1107	1.1125	1.0787	1.1110	1.1135	1.1185	1.1200	1.1191	1.1193	1.1213
		θ	Avg	0.4749	0.5301	0.5294	0.5298	0.5391	0.5267	0.5096	0.5089	0.5094	0.5084	0.5056
			RE	1.0832	1.1657	1.1650	1.1655	1.1620	1.1583	1.0930	1.0917	1.0926	1.0877	1.0831
	40	λ	Avg	1.6498	1.7053	1.7007	1.6987	1.6970	1.6962	1.7548	1.7487	1.7528	1.7505	1.7428
			RE	1.0005	1.0557	1.0567	1.0505	1.0554	1.0568	1.0581	1.0600	1.0587	1.0591	1.0613
		θ	Avg	0.5134	0.5354	0.5347	0.5352	0.5344	0.5322	0.5315	0.5307	0.5312	0.5303	0.5275
			RE	1.0942	1.2171	1.2169	1.2117	1.2073	1.2083	1.2507	1.2504	1.2506	1.2422	1.2405
	45	λ	Avg	1.6043	1.7907	1.7823	1.7878	1.7849	1.7748	1.8252	1.8166	1.8223	1.8192	1.8088
			RE	1.0004	1.0521	1.0541	1.0528	1.0513	1.0534	1.0502	1.0519	1.0508	1.0514	1.0533
		θ	Avg	0.4829	0.5113	0.5569	0.5111	0.5104	0.5542	0.4841	0.5322	0.4838	0.4831	0.5291
			RE	1.0912	1.1480	1.2292	1.1478	1.1256	1.2247	1.0595	1.2014	1.0588	1.0587	1.1911
100	85	λ	Avg	1.5376	1.6560	1.6535	1.6552	1.6542	1.6510	1.6899	1.6870	1.6889	1.6878	1.6841
			RE	1.0229	1.0105	1.0107	1.0106	1.0104	1.0106	1.0161	1.0164	1.0162	1.0163	1.0168
		θ	Avg	0.4783	0.5384	0.5382	0.5383	0.5382	0.5376	0.5512	0.5307	0.5511	0.5540	0.5300
			RE	1.0093	1.1711	1.1712	1.1712	1.1685	1.1695	1.2338	1.2269	1.2339	1.2291	1.2208
	90	λ	Avg	1.5275	1.7079	1.7043	1.7067	1.7054	1.7008	1.7311	1.7273	1.7298	1.7284	1.7235
			RE	1.0594	1.0087	1.0091	1.0088	1.0084	1.0088	1.0064	1.0066	1.0065	1.0062	1.0065
		θ	Avg	0.4512	0.5330	0.5328	0.5329	0.5328	0.5322	0.5434	0.5319	0.5433	0.5431	0.5311
			RE	1.0102	1.1799	1.1802	1.1800	1.1757	1.1776	1.1616	1.1550	1.1617	1.1584	1.1516
	95	λ	Avg	1.0598	1.7642	1.7599	1.7628	1.7612	1.7557	1.7808	1.7763	1.7793	1.7776	1.7718
			RE	1.0865	1.0243	1.0246	1.0244	1.0244	1.0248	1.0143	1.0148	1.0145	1.0141	1.0147
		θ	Avg	0.5444	0.5178	0.5178	0.5179	0.5177	0.5172	0.5256	0.5254	0.5255	0.5254	0.5248
			RE	1.0101	1.1328	1.1322	1.1329	1.1233	1.1259	1.1469	1.1473	1.1470	1.1401	1.1426

Table 11. Best unbiased, conditional median, and Bayes predictive estimates of $x_{r+1}$.

n	r	Classical		Bayesian prediction under NIN Prior					Bayesian prediction under IN Prior
		prediction		Linex			GEL		Linex			GEL
		BUP	CMP	$h=$−0.25	$h=$0.5	SEL	$q=$−0.5	$q=$1.25	$h=$−0.25	$h=$0.5	SEL	$q=$−0.5	$q=$1.25
30	20	5.7944	1.8132	14.697	4.6531	4.9345	13.447	4.6629	9.2793	8.3663	8.9371	8.8668	8.6234
	23	3.4946	1.6188	8.3400	7.3005	7.9380	7.8487	7.5431	5.5707	5.1712	5.4267	5.3770	5.2044
	25	2.1181	1.4229	5.0982	4.6531	4.9345	4.8743	4.6629	3.4384	3.2393	3.3685	3.3293	3.1916
50	35	8.3295	2.0559	19.280	16.642	18.342	18.246	17.896	14.372	13.103	13.921	13.859	13.639
	40	4.6004	1.8598	10.230	9.3457	9.9277	9.8680	9.6518	7.8908	7.3806	7.7117	7.6671	7.5088
	45	1.5788	1.3897	3.5455	3.3695	3.4849	3.4514	3.3315	2.8088	2.6957	2.7700	2.7425	2.6440
100	85	5.9698	2.1180	12.269	11.680	12.078	12.045	11.927	10.782	10.236	10.597	10.562	10.437
	90	3.0980	1.8245	6.4534	6.2097	6.3705	6.3443	6.2507	5.7943	5.5670	5.7174	5.6900	5.5928
	95	0.9605	1.2959	2.2568	2.2034	2.2389	2.2227	2.1642	1.9813	1.9420	1.9679	1.9545	1.9088

Table 12. Predictive interval estimates of $x_{r+1}$.

n	r	Classical prediction		Bayesian prediction
		Pivotal/AIL	HCD/AIL	NIN prior/AIL	IN prior/AIL
30	20	(0.8878–3.7304)/2.8426	(0.9980–1.8486)/0.8506	(1.2809–5.7394)/4.4585	(4.7664–8.9371)/4.1707
	23	(0.6837–3.6368)/2.9531	(0.7911–2.3530)/1.5619	(5.7038–8.9037)/3.1999	(4.0944–5.3029)/1.2085
	25	(0.5005–3.5631)/3.0626	(0.6008–2.8995)/2.2987	(1.2809–4.9790)/3.6981	(1.2075–3.8513)/2.6438
50	35	(1.1049–3.9891)/2.8842	(1.2202–2.0042)/0.7840	(15.385–18.342)/2.9570	(11.880–13.921)/2.0410
	40	(0.8644–4.6810)/3.8166	(0.9810–2.6554)/1.6744	(7.8975–10.839)/2.9424	(6.5920–7.5108)/0.9188
	45	(0.4573–4.4898)/4.0325	(0.5556–3.6798)/3.1242	(1.4826–3.7862)/2.3036	(1.4513–2.9796)/1.5283
100	85	(1.0722–6.3059)/5.2337	(1.1966–3.0572)/1.8606	(10.674–13.454)/2.7800	(9.2148–11.374)/2.1592
	90	(0.8045–4.5719)/3.7674	(0.9215–3.6986)/2.7771	(5.3632–6.9621)/1.5989	(5.7174–6.0048)/0.2874
	95	(0.4283–3.3740)/2.9457	(0.5133–3.6145)/3.1012	(1.7146–3.2446)/1.5300	(1.7479–2.2687)/0.5308

8. Conclusion

In this article, we considered the problem of estimating the unknown parameters of Poisson-exponential distribution under type-II censoring, and for this purpose both the classical and Bayesian estimators have been taken into consideration. We observed that MLEs of the unknown parameters of the distribution do not admit closed form; therefore, we employed the EM algorithm to compute the maximum likelihood estimates, and further computed approximate confidence intervals using the idea of missing information principle and the asymptotic normality of the MLEs. In the Bayesian approach, we obtained Bayes estimators under SEL, LINEX and GEL loss functions using Lindley and importance sampling as the Bayes estimators were not in the closed form. Further, with the help of importance sampling and the form of posterior density, we also computed highest posterior density interval estimates using the method of Chen and Shao (1999). Performance of EM algorithm over NR and Bayes estimators under informative prior have been found more satisfactory in our simulation study on the basis of average and mean square error values. Furthermore, to provide predictive inference on the censored observation we made use of best unbiased and conditional median predictors under classical approach and importance sampling under Bayesian approach. In real data analysis, we observed that the predictive observation are close to the true observations and further the predictive interval estimates contain the true observations. Finally, we mention that the statistical inference on PE distribution under generalization of type-II, progressive or progressive hybrid censoring schemes can be seen as future direction.

Acknowledgement

The authors would like to thank the Editor, associate Editor, and anonymous reviewers for the constructive and valuable suggestions which led to the improvement in an earlier version of this article.

Disclosure Statement

No potential conflict of interest was reported by the authors.

Appendix

For the two parameter case $({\Theta }_1,{\Theta }_2),$ the Lindley’s approximation to $E(g(\varphi )|\mathit{x})$ is given by, (15) $$\widehat{g}(\varphi )=g(\widehat{{\Theta }_1},\widehat{{\Theta }_2})+\frac{1}{2}[A+l_{30}{B}_{12}+l_{03}{B}_{21}+l_{21}{C}_{12}+l_{12}{C}_{21}]+{\rho }_1{A}_{12}+{\rho }_2{A}_{21},$$(15) where, $$\begin{array}{c}A=\sum _{i=1}^2\sum _{j=1}^2{w}_{ij}{\sigma }_{ij},\hspace{1em}\hspace{1em}\hspace{1em}{l}_{ij}=\frac{{\partial }^{i+j}L({\Theta }_1,{\Theta }_2)}{\partial {\Theta }_1^i\partial {\Theta }_2^j},\\ {\rho }_i=\frac{\partial \rho }{\partial {\Theta }_i},\hspace{1em}\hspace{1em}\hspace{1em}{w}_i=\frac{\partial g}{\partial {\Theta }_i},\hspace{1em}\hspace{1em}\hspace{1em}{w}_{ij}=\frac{{\partial }^{2}g}{\partial {\Theta }_i\partial {\Theta }_j},\\ \rho =\hspace{0.17em}\text{ln}\hspace{0.17em}\pi ({\Theta }_1,{\Theta }_2),\hspace{1em}\hspace{1em}\hspace{1em}{A}_{ij}=w_i{\sigma }_{ii}+w_j{\sigma }_{ji},\\ B_{ij}=(w_i{\sigma }_{ii}+w_j{\sigma }_{ij}){\sigma }_{ii},\hspace{1em}\hspace{1em}\hspace{1em}{C}_{ij}=3w_i{\sigma }_{ii}{\sigma }_{ij}+w_j({\sigma }_{ii}{\sigma }_{jj}+2{\sigma }_{ij}^2).\end{array}$$

Here, $L(.,.)$ denotes the likelihood function, $\pi ({\Theta }_1,{\Theta }_2)$ denotes the prior distribution, and σ_ij is the $(i,j)th$ element of the inverse of the Fisher information matrix. Note, that the expressions in (15)(15) $$\widehat{g}(\varphi )=g(\widehat{{\Theta }_1},\widehat{{\Theta }_2})+\frac{1}{2}[A+l_{30}{B}_{12}+l_{03}{B}_{21}+l_{21}{C}_{12}+l_{12}{C}_{21}]+{\rho }_1{A}_{12}+{\rho }_2{A}_{21},$$(15) are evaluated at the maximum likelihood estimates $({\widehat{\Theta }}_1,{\widehat{\Theta }}_2)$. For the case of our estimation problem with $({\Theta }_1,{\Theta }_2)=(\theta ,\lambda )$, the approximate Bayes estimates of θ and λ are computed using the expression (15)(15) $$\widehat{g}(\varphi )=g(\widehat{{\Theta }_1},\widehat{{\Theta }_2})+\frac{1}{2}[A+l_{30}{B}_{12}+l_{03}{B}_{21}+l_{21}{C}_{12}+l_{12}{C}_{21}]+{\rho }_1{A}_{12}+{\rho }_2{A}_{21},$$(15) , and the involved expressions are evaluated as, $$\begin{array}{c}{l}_{11}={\displaystyle \sum _{i=1}^r{x}_i}{e}^{-\lambda x_i}-(n-r)[\frac{x_r{e}^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}[1-\theta e^{-\lambda x_r}]}{1-e^{-\theta e^{-\lambda x_r}}}-\frac{\theta x_r{e}^{-2\lambda x_r}{e}^{-2\theta e^{-\lambda x_r}}}{{(1-e^{-\theta e^{-\lambda x_r}})}^2}],\\ l_{02}=-\frac{r}{{\lambda }^2}-\theta {\displaystyle \sum _{i=1}^r{x}_i^2}{e}^{-\lambda x_i}+(n-r)[\frac{\theta x_r^2{e}^{-\lambda x_r-\theta e^{-\lambda x_r}}(1-\theta )}{1-e^{-\theta e^{-\lambda x_r}}}-\frac{{\theta }^2{x}_r^2{e}^{-2\lambda x_r-2\theta e^{-\lambda x_r}}}{{(1-e^{-\theta e^{-\lambda x_r}})}^2}],\\ l_{20}=-\frac{r}{{\theta }^2}+\frac{r{e}^{-\theta }}{1-e^{-\theta }}+\frac{r{e}^{-2\theta }}{{(1-e^{-\theta })}^2}+(n-r)\{\frac{(1-e^{-\theta })}{1-e^{-\theta e^{-\lambda x_r}}}\\ \times [-\frac{e^{-2\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}-\frac{2e^{-\lambda x_r-\theta }{e}^{-\theta e^{-\lambda x_r}}}{{(1-e^{-\theta })}^2}+\frac{2e^{-2\theta }(1-e^{-\theta e^{-\lambda x_r}})}{{(1-e^{-\theta })}^3}+\frac{e^{-\theta }(1-e^{-\theta e^{-\lambda x_r}})}{{(1-e^{-\theta })}^2}]\\ -\frac{1}{{(1-e^{-\theta e^{-\lambda x_r}})}^2}[e^{-2\lambda x_r-2\theta e^{-\lambda x_r}}-\frac{(1-e^{-\theta e^{-\lambda x_r}})e^{-\lambda x_r-\theta }{e}^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}]\\ +\frac{e^{-\theta }}{1-e^{-\theta e^{-\lambda x_r}}}[\frac{e^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}-\frac{e^{-\theta }(1-e^{-\theta e^{-\lambda x_r}})}{{(1-e^{-\theta })}^2}]\},\\ l_{12}=-{\displaystyle \sum _{i=1}^r{x}_i^2}{e}^{-\lambda x_i}+(n-r)[\frac{x_r^2{e}^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}[{\theta }^2{e}^{-2\lambda x_r}-3\theta e^{-\lambda x_r}+1]}{1-e^{-\theta e^{-\lambda x_r}}}\\ +\frac{3\theta x_r^2{e}^{-2\lambda x_r}{e}^{-2\theta e^{-\lambda x_r}}(\theta -1)}{{(1-e^{-\theta e^{-\lambda x_r}})}^2}+\frac{2{\theta }^2{x}_r^2{e}^{-3\lambda x_r}{e}^{-3\theta e^{-\lambda x_r}}}{{(1-e^{-\theta e^{-\lambda x_r}})}^3}],\\ l_{21}=(n-r)[\frac{x_r{e}^{-2\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}[2-\theta ]}{1-e^{-\theta e^{-\lambda x_r}}}+\frac{x_r{e}^{-2\lambda x_r}{e}^{-2\theta e^{-\lambda x_r}}[2-3\theta e^{-\lambda x_r}]}{{(1-e^{-\theta e^{-\lambda x_r}})}^2}-\frac{2\theta x_r{e}^{-3\lambda x_r}{e}^{-3\theta e^{-\lambda x_r}}}{{(1-e^{-\theta e^{-\lambda x_r}})}^3}],\end{array}$$ $$\begin{array}{c}{l}_{30}=\frac{2r}{{\theta }^3}-\frac{r{e}^{-\theta }}{1-e^{-\theta }}-\frac{3r{e}^{-2\theta }}{{(1-e^{-\theta })}^2}-\frac{2r{e}^{-3\theta }}{{(1-e^{-\theta })}^3}\\ +\frac{1}{1-e^{-\theta e^{-\lambda x_r}}}[(n-r)\{\frac{e^{-3\lambda x_r}{e}^{-\theta e^{-\theta e^{-\lambda x_r}}}}{1-e^{-\theta }}+\frac{3e^{-2\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}{e}^{-\theta }}{{(1-e^{-\theta })}^2}+\frac{6e^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}{e}^{-2\theta }}{{(1-e^{-\theta })}^3}\\ +\frac{3e^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}{e}^{-\theta }}{{(1-e^{-\theta })}^2}-\frac{6(1-e^{-\theta e^{-\lambda x_r}})e^{-3\theta }}{{(1-e^{-\theta })}^4}-\frac{6(1-e^{-\theta e^{-\lambda x_r}})e^{-2\theta }}{{(1-e^{-\theta })}^3}\\ -\frac{(1-e^{-\theta e^{-\lambda x_r}})e^{-\theta }}{{(1-e^{-\theta })}^2}\}(1-e^{-\theta })]-\frac{1}{{(1-e^{-\theta e^{-\lambda x_r}})}^2}[2(n-r)\{-\frac{e^{-2\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}\\ -\frac{2e^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}{e}^{-\theta }}{{(1-e^{-\theta })}^2}+\frac{2(1-e^{-\theta e^{-\lambda x_r}})e^{-2\theta }}{{(1-e^{-\theta })}^3}+\frac{(1-e^{-\theta e^{-\lambda x_r}})e^{-\theta }}{{(1-e^{-\theta })}^2}\}(1-e^{-\theta })e^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}]\\ +\frac{1}{(1-e^{-\theta e^{-\lambda x_r}})}[2(n-r)\{-\frac{e^{-2\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}-\frac{2e^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}{e}^{-\theta }}{{(1-e^{-\theta })}^2}+\frac{2(1-e^{-\theta e^{-\lambda x_r}})e^{-2\theta }}{{(1-e^{-\theta })}^3}\\ +\frac{(1-e^{-\theta e^{-\lambda x_r}})e^{-\theta }}{{(1-e^{-\theta })}^2}\}{e}^{-\theta }]+2(n-r)\{[\frac{e^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}-\frac{(1-e^{-\theta e^{-\lambda x_r}})e^{-\theta }}{{(1-e^{-\theta })}^2}]\\ \times \frac{(1-e^{-\theta })e^{-2\lambda x_r}{e}^{-2\theta e^{-\lambda x_r}}}{{(1-e^{-\theta e^{-\lambda x_r}})}^3}-\frac{2[\frac{e^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}-\frac{(1-e^{-\theta e^{-\lambda x_r}})e^{-\theta }}{{(1-e^{-\theta })}^2}]e^{-\theta -\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}}{{(1-e^{-\theta e^{-\lambda x_r}})}^2}\\ +\frac{[\frac{e^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta }}-\frac{(1-e^{-\theta e^{-\lambda x_r}})e^{-\theta }}{{(1-e^{-\theta })}^2}](1-e^{-\theta })e^{-2\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}}{1-e^{-\theta e^{-\lambda x_r}}}\}\\ l_{03}=\frac{3r}{{\lambda }^3}+\theta {\displaystyle \sum _{i=1}^r{x}_i^3}{e}^{-\lambda x_i}-(n-r)[\frac{\theta x_r^3{e}^{-\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}[3\theta e^{-\lambda x_r}-{\theta }^2{e}^{-2\lambda x_r}+1]}{1-e^{-\theta e^{-\lambda x_r}}}\\ +\frac{3{\theta }^2{x}_r^3{e}^{-2\lambda x_r}{e}^{-\theta e^{-\lambda x_r}}[1-\theta e^{-\lambda x_r}]}{1-e^{-\theta e^{-\lambda x_r}}}-\frac{2{\theta }^3{x}_r^3{e}^{-3\lambda x_r}{e}^{-3\theta e^{-\lambda x_r}}}{{(1-e^{-\theta e^{-\lambda x_r}})}^3}].\end{array}$$