Empirical Bayes Estimation for Exponential Model Using Non-parameter Polynomial Density Estimator

Manfeng Liu,Haiping Ren

doi:10.19026/rjaset.5.4964

Abstract

In this study, we study the empirical Bayes estimation of the parameter of the exponential distribution. In the empirical Bayes procedure, we employ the non-parameter polynomial density estimator to the estimation of the unknown marginal probability density function, instead of estimating the unknown prior probability density function of the parameter. Empirical Bayes estimators are derived for the parameter of the exponential distribution under squared error and LINEX loss functions. We use numerical examples to compare the empirical Bayes estimators we obtained under squared error and LINEX loss functions and we get the result of the mean square error of the empirical Bayes estimator under LINEX loss is usually smaller than the estimator under squared error loss function, so it is more better.

Highlights

The exponential distribution is one of the most important distributions in life-testing and reliability studies
If the squared error loss function is used for each possible value of θ, the Bayes estimator for θ is defined as the posterior mean of θ given z = Z(X), that is:
Theorem: In the following discussion, we always suppose that X = (X1, X2,..., Xc) is a random sample with size n which is drawn from exponential distribution (1), let z be a value of the statistic Z(X)

Summary

Introduction

The exponential distribution is one of the most important distributions in life-testing and reliability studies. We will employ empirical Bayes method to the estimation of the parameter of exponential distribution. The Bayes estimator of θ, denoted by under the LINEX loss function is the value which minimizes (4), it is:

Results

Conclusion