Bayesian Estimation for Two Parameters of Exponential Distribution under Different Loss Functions

Abstract

In this paper, two parameters for the Exponential distribution were estimated using theBayesian estimation method under three different loss functions: the Squared error loss function,the Precautionary loss function, and the Entropy loss function. The Exponential distribution priorand Gamma distribution have been assumed as the priors of the scale γ and location δ parametersrespectively. In Bayesian estimation, Maximum likelihood estimators have been used as the initialestimators, and the Tierney-Kadane approximation has been used effectively. Based on the MonteCarlosimulation method, those estimators were compared depending on the mean squared errors (MSEs).The results showed that the Bayesian estimation under the Entropy loss function,assuming Exponential distribution and Gamma distribution priors for the scale and locationparameters, respectively, is the best estimator for the scale parameter. The best estimation methodfor location is the Bayesian estimation under the Entropy loss function in case of a small value ofthe scale γ (say γ < 1). Bayesian estimation under the Precautionary loss function is the best incase of a relatively large value of the scale γ (say γ > 1).