A Bayesian Framework for Estimating the Shape Parameter of an Exponential Poisson-lindley Distribution

Abstract

A Bayesian analysis of the shape parameter of an Exponential Poisson Lindley Distribution (ExPLinD) is presented in this study. This study examined the Bayesian estimation of the shape parameters of an ExPLinD using both informative and non-informative priors. Uniform and Jeffrey Priors were used as informative priors, while a gamma prior was adopted as a non-informative prior. These priors were combined with different error loss functions (Squared Error Loss Function (SELF), Precautionary Loss Function (PLF) and Quadratic Loss Function (QLF)) to allow for the possibility of different combinations and scenarios of prior and loss function that produced the best estimate of the shape parameter of an ExPLinD. Simulation study was conducted using Mean Squared Errors (MSE) as a metric, the Quadratic loss function produced the best estimator of the shape parameter of an ExPLinD compared to estimates from the Maximum Likelihood Estimation (MLE), SELF and PLF having the lowest estimated value of this metric except for when α is increased from 0.7 to 3.7 and “a” is increased from 1.0 to 3.5, where the PLF under Gamma prior presented less MSE than the QLF. The result also revealed that the values of the other parameters have no effect on the estimators of the shape parameter, as changing the values of these other parameters alone does not affect the MSE. This makes the QLF the best loss function to be used in obtaining a Bayesian estimate of an ExPLinD, as it will have minimal influence on inference to be made.