Objective Prior Distributions to Estimate the Parameters of the Poisson-Exponential Distribution

In this paper, a set of important objective priors are examined for the Bayesian estimation of the parameters present in the Poisson-Exponential distribution P E. We derived the multivariate Jeffreys prior and the Maximal Data Information Prior. Reference prior and others priors proposed in the literature are also analyzed. We show that the posterior densities resulting from these approaches are proper although the respective priors are improper. Monte Carlo simulations are used to compare the efficiencies and to assess the sensitivity of the choice of the priors, mainly for small sample sizes. This simulation study shows that the mean square error, mean bias and coverage probability of credible intervals under Gamma, Jeffreys' rule and Box & Tiao priors presented equal results, whereas Jeffreys and Reference priors showed the best results. The MDIP prior had a worse performance in all analyzed situations showing not to be indicated for Bayesian analysis of the P E distribution. A real data set is analyzed for illustrative purpose of the Bayesian approaches.