Analysis of parameters of the exponentiated inverse Rayleigh distribution under the Bayesian framework

Abstract

Estimating the unknown parameter(s) of distribution using Bayesian framework is a core topic in statistical literature. This study focuses on the Bayesian estimation and prior selection for the scale and shape parameters of the exponentiated inverse Rayleigh distribution. We consider both informative (chi-square, inverse Lévy) and non-informative (uniform, Jeffreys) priors to update the current state of knowledge regarding the unknown parameters. The squared error loss function (SELF), LINEX loss function (LLF), precautionary loss function (PLF), and quasi-quadratic loss function (QQLF) are employed to demonstrate the effectiveness of priors while estimating the parameters. Expressions for posterior distributions, Bayes estimators (BE), Bayes posterior risks (BPR), credible intervals, and predictive intervals are derived under the aforementioned conditions. Extensive simulation as well as real data analysis is carried out to show the relative performances of the priors and loss functions by comparing the respective BPRs. The results reveal that the inverse Lévy prior outperforms the other priors in terms of minimum BPR and providing tighter credible and predictive intervals while estimating the scale parameter. Whereas, for the shape parameter, the gamma prior shows superior performance. The real data analysis cements the findings of the simulation study.