On Bayesian Estimation of Loss and Risk Functions

Randhir Singh

doi:10.11648/j.sjams.20210903.11

Abstract

Loss functions and Risk functions play very important role in Bayesian estimation. This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the H(r, theta), (theta being the unknown parameter) distribution The estimation has been performed under Rukhin’s loss function. The importance of this distribution is that it contains some important distributions such as the Half Normal distribution, Rayleigh distribution and Maxwell’s distribution as particular cases. The inverse Gamma distribution has assumed as the prior distribution for the unknown parameter theta. This prior distribution is a Natural Conjugate prior distribution for the unknown parameter because the posterior probability density function of the unknown parameter is also inverse gamma distribution The Rukhin’s loss function involves another loss function denoted by w(theta, delta) he form of w(theta, delta) is important as it changes the estimate. In this paper, three forms of w(theta, delta) have been taken and corresponding estimates have been derived. The three, forms are, the Squared Error Loss Function (SELF) and two different forms of Weighted Squared Error Loss Function (WSELF) namely, the Minimum Expected Loss (MELO) Function and the Exponentially Weighted Minimum Expected Loss (EWMELO) Function have been considered. A criterion of performance of various form of w(theta, delta) has ben defined. It has been proved that among three forms of w(theta, delta), considered here, the form corresponding to EWMELO is most dominant.

Highlights

A continuous random variable X is said to have H(r,θ distribution, if its probability density function is given by, f x, θ, if x 0, θ 0, r 0 (1) 0, Otherwise.This distribution covers some important distribution for various values of r
Squared Error Loss Function (SELF) and two forms of Weighted Squared Error Loss Function (WSELF) was used by Singh [7,8] in the study of reliability of a multicomponent system and Bayesian Estimation of the mean and distribution function of Maxwell’s distribution under the assumption of conjugate prior distribution
Guobing Fan [9] has derived the Bayes estimator of the loss and risk function of Maxwell’s distribution using inverse Gamma distribution as the prior distribution for θ and squared error loss function under the criterion of loss function proposed by Rukhin [10], which is given by

Summary

Introduction

A continuous random variable X is said to have H(r,θ distribution, if its probability density function is given by, f x, θ. Tyagi and Bhattacharya [3,4,5] studied the classical and Bayesian estimation of Rayleigh and Maxwell’s distribution They used only SELF for Bayesian estimation. SELF and two forms of WSELF was used by Singh [7,8] in the study of reliability of a multicomponent system and Bayesian Estimation of the mean and distribution function of Maxwell’s distribution under the assumption of conjugate prior distribution. Day and Sudhanshu obtained Bayes estimators of parameters of Maxwell distribution bu using non-informative as well as conjugate prior distributions. They used quadratic loss function, SELF and MLINEX loss function. Have recently considered Bayesian estimation for Exponentiated Inverted Weibull distribution an Logarithmic Transformed Exponential distribution under different loss functions

Basic Concepts

Bayesian Estimation of Loss and Risk Function

Conclusion