Admissibility Estimation of Burr Type XI Distribution Under Entropy Loss Function Based on Record Values

The aim of this paper is to study the estimation of parameter of Burr Type XI distribution on the basis of lower record values. First, the minimum variance unbiased estimator and maximum likelihood estimator are obtained. Then the Bayes and empirical Bayes estimators of the unknown parameter are derived under entropy loss function. Finally, the admissibility and inadmissibility of a class of inverse linear estimators are discussed.


Introduction
Burr distribution contains twelve forms, which first introduced by Burr in 1942 [1]. However, Burr type XII distribution is the most of the authors have considered [2][3][4][5][6], But other types of Burr distribution do not get much attention. Feroze and Aslam [7] studied the Bayes estimation of the parameter of Burr type XI distribution under the assumption of eight priors (informative, non-informative, single and mixture of priors). The entropy and precautionary loss functions were used in their Bayesian analysis. Feroze et al. [8] studied the Bayes estimation and prediction of the Burr type XI distribution based on censored samples on the basis of Five informative and non-informative priors under five different (symmetric and asymmetric) loss functions for posterior analysis.
This paper is devoted to study the Bayes estimation problem of the unknown parameter of the Burr type XI distribution with cumulative distribution function and probability density function given as follows, respectively (Feroze and Aslam [7]): Here 0 θ > is the unknown parameter, ∝ denotes the probability density function omits the function of a proportionality constant with respect to the parameter θ .
Record values is an important order statistics, which was first studied by Chandler in 1952 [9]. It has many application in various fields, such as weather, sports, economics, lifetests, stock market and so on, and the statistical inference study of record values has received great attention [10][11][12][13]. For example, Wang et al. [14] investigated the point estimation and confidence intervals estimation for the parameter of Weibull distribution based on record values. Arabi Belaghi et al. [15] considered several different types confidence intervals for the scale parameter of Burr type XII distribution based on upper record values. They put forward improved confidence intervals estimation based on the preliminary test technique, Thompson shrinkage technique and Bayesian approach. Barranco-Chamorro et al. [16] proposed two new maximum likelihood methods for estimating the unknown sample size in a simple random sample from an absolutely continuous population one the basis of record values.
The admissibility of estimators is an important topic in statistical inference. Many authors studied the admissibility and inadmissibility of various estimators for different distributions. For example, Wen and Levy [17] considered properties of Bayes estimators in a normal mean problem derived under a bounded loss function called BLINEX Loss function. Zakerzadeh and Zahraie [18] considered the dmissibility of estyimators in non-regular family under squared-log error loss function. Cao and kong [19] studied the general admissibility for a class of linear estimators in a general multivariate linear model under balanced loss function. More relevant reference can be found in [20][21][22].
The purpose of this paper is to study the estimation and admissible characters of estimators of the parameter of Burr type XI distribution.

Preliminary Knowledge
This section will first recall some definitions of record values. Then derive the maximum likelihood estimator (MLE) and minimum variance unbiased estimator of the parameter of Burr Type XI distribution. Definition 1. Let 1 2 , , X X ⋯ be a sequence of independent and identically distributed random variables. Suppose Then the likelihood function of X is given as follows (Arnold et al.
is an unbiased estimator of θ , and it is also a function the complete and sufficient statistics T . Therefore, by using Lehmann and Scheffe's theorem, the estimator T n 1 − is the minimum variance unbiased estimator of θ ,

Bayes Estimation
Loss function is an important aspect in Bayesian analysis. This section will derive the Bayes estimatior under a useful asymmetric loss function named entropy loss function (Dey et al. [24]): The Bayes estimator under the entropy loss is denoted by BE θˆ, given by Suppose the prior distribution of parameter θ is Gamma According to equations (3) and (10), using Bayesian theorem, we can get the posterior probability density function as follows: and by equation (9), the Bayes estimator of θ under entropy loss function is which is of the form In the rest of this section, all obtained estimators will be compared on the basis of their risks under the entropy loss function. We will also study the conditions under which general inverse linear estimators are admissible in terms of risk function.
then the conclusion is proved. The Bayes estimator with respect to ( ) k π θ under the entropy loss function can be derived as in (11)

Conclusion
Admissibility of estimators is an important topic in statistical inference field. This paper considers the estimation of the unknown parameter from the Burr Type are studied under different conditions. As a result, the minimum variance unbiased estimator is admissible, but the empirical Bayes estimator and the maximum likelihood estimator are inadmissible