Bayesian Inference for the Gamma Zero-Truncated Poisson Distribution with an Application to Real Data

Abstract

This article presents Bayesian estimation methods applied to the gamma zero-truncated Poisson (GZTP) and the complementary gamma zero-truncated Poisson (CGZTP) distributions, encompassing both one-parameter and two-parameter models. These distributions are notably flexible and useful for modeling lifetime data. In the one-parameter model case, the Jeffreys prior is mathematically derived. The use of informative and noninformative priors, combined with the random walk Metropolis algorithm within a Bayesian framework, generates samples from the posterior distributions. Bayesian estimators’ effectiveness is examined through extensive simulation studies, in comparison with the maximum likelihood method. Results indicate that Bayesian estimators provide more precise parameter estimates, even with smaller sample sizes. Furthermore, the study and comparison of the coverage probabilities (CPs) and average lengths (ALs) of the credible intervals with those from Wald intervals suggest that Bayesian credible intervals typically yield shorter ALs and higher CPs, thereby demonstrating the effectiveness of Bayesian inference in the context of GZTP and CGZTP distributions. Lastly, Bayesian inference is applied to real data.

Keywords

• gamma zero-truncated Poisson
• complementary gamma zero-truncated Poisson
• random walk metropolis
• squared error loss
• Bayesian estimation
• credible interval