Admissibility Estimation of Pareto Distribution Under Entropy Loss Function Based on Progressive Type-II Censored Sample

The aim of this paper is to study the estimation of Pareto distribution on the basis of progressive type-II censored sample. First, the maximum likelihood estimator (MLE) is derived. Then the Bayes estimator of the unknown parameter of Pareto distribution is derived on the basis of Gamma prior distribution under entropy loss function. Further the empirical Bayes estimator also obtained by using maximum likelihood on the basis of Bayes estimator. Finally, the admissibility of a class of inverse linear estimators are discussed under suitable conditions.


Introduction
In practical life testing experiments, considering with the time limitation and/or other restrictions (such as cost, material resources, etc.), the experimenter may not always be in a position to observe the life times of all the products in a lifetime test [1,2]. Censored samples often arise in practice. Progressive Type II censored sampling test is one of the most important method of obtaining data in lifetime researches. The statistical inference studies for various distributions when sample belongs to progressive censoring have attracted many authors' attention. For example, Ng et al. [3] computed the expected Fisher information and the asymptotic variancecovariance matrix of the ML estimates based on progressively Type-II censored sample from Weibull distribution They also discussed the construction of progressively censored reliability sampling plans. Soliman et al. [4] investigated the point and interval estimations for the modified Weibull distribution based on progressively type-II censored sample. Yang [5] derived the maximum likelihood estimation of Weibull distribution under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial distribution.Wu [6] considered the estimation problem of the two-parameter bathtub-shaped lifetime distribution based a progressively type-II censored sample. Cho et al. [7] discussed the Bayes estimation of the entropy of a twoparameter Weibull distribution based on the generalized progressively censored sample. Bhattacharya et al. [8] proposed an optimum life-testing plans under Type-II progressive censoring scheme using variable neighborhood search algorithm. Laumen and Cramer [9] discussed the likelihood inference and statistical test procedure for the lifetime performance index in the presence of progressive censoring. Khorram and Farahani [10] considered the maximum likelihood estimation and bayes estimation of parameters of weighted exponential distribution based on progressively Type-II censored sample.
Many statistical inference problems have been discussed for various lifetime distributions, such as exponential distribution, Weibull distribution, etc. Wu and Chang [11] pointed out that Pareto distribution can be regarded as a suitable alternative distribution in modeling product's lifetime. Statistical inference about Pareto distribution receives great attention by authors in recent yeasrs. For example, Raqab et al. [12] derived the best linear unbiased Type-II Censored Sample predictors, maximum likelihood predictors and approximate maximum likelihood predictors of times to failure of units censored from Pareto distribution. Kulldorff and Vannman [13] obtained the best linear unbiased estimates based on the complete sample and the asymptotically best linear unbiased estimates based on a few selected order statistics. Fu et al. [14] discussed the Bayesian estimation of Pareto distributions under progressive Type-II censoring on the basis of several types of noninformative priors, i.e. Jeffreys prior, two reference priors and two general forms of second order probability matching prior. Saldaña-Zepeda et al. [15] proposed a goodness of fit test procedure for the Pareto distribution, when the observations drawn from Type II right censoring.
Assume that the repair time X follows the Pareto distribution with the following probability density function (pdf) and cumulative distribution function (cdf) respectively: This paper will discuss the Bayes estimation of the parameter of Pareto distribution under entropy loss function on the basis of progressive type-II censored sample. The admissibility of estimators is an important topic [16][17][18][19].
Thus this paper will also study the admissibility and inadmissibility of a class of inverse linear estimators under suitable conditions. The remains of this paper are organized as follows. The MLE and Bayesian estimators of the parameter are obtained in Section 2. In Section 3, the admissibility and inadmissibility of estimators with inverse linear form are discussed. A conclusion is finally made in Section 4.

Preliminary Knowledge
The progressively Type II censoring scheme can be described as follows [20].
First, the experimenter places n units or individual on test.
(i) When the first failure is observed at 1: : m n X , then randomly select 1 r surviving unites and remove them.
⋯ is a progressively Type-II censored sample from a life test on n items whose lifetimes follows Pareto distribution with pdf shown in (1), 1: : 2: : : : is the corresponding observation of X and 1 2 , , , m r r r ⋯ denote the corresponding numbers of units removed from the test.

Maximum Likelihood Estimation
The likelihood function of θ under given progressively type-II censored sample 1: : 2: : : : Here : : f x θ and ( ; ) F x θ are given respectively by (1) and (2), and (1) and (2) into (3), the likelihood function is given by The natural logarithm of likelihood function is given by Then the MLE of θ can is the solution of the following Thus the MLE of θ can be easily solved as

Bayes Estimation
This subsection will discuss the Bayes estimation of the parameter θ of Pareto distribution (1) based on progressively Type-II censored sample under the following entropy loss function:  ⋯ is a progressively Type-II censored sample from a life test on n items whose lifetimes have Pareto distribution (1), and 1 2 , , , m r r r ⋯ denote the corresponding numbers of units removed from the test.
Note that the MLE, Bayes estimator and empirical estimators obtained in Section 2 are all the special cases of a class of inverse linear estimators of the form ( 1)( ) Viveros & Balakrishnan [23] showed that the generalized spacings 1 2 , , , m Z Z Z ⋯ are all independent and identically distributed (i.i.d.) as standard exponential with mean 1. In fact,

Conclusions
This paper considers the estimation of the unknown parameter of Pareto distribution based on progressively type II censored samples. The MLE, Bayes estimator and empirical Bayes estimators are obtained. These estimators all belong to a class of inverse linear estimators with the form