Bayesian and classical inference for the generalized gamma distribution and related models

Abstract

The generalized gamma (GG) distribution is an important model that has proven to be very flexible in practice for modeling data from several areas. This model has important sub-models, such as the Weibull, gamma, lognormal, Nakagami-m distributions, among others. In this work, our main objective is to develop different estimation procedures for the unknown parameters of the generalized gamma distribution and related models (Nakagami-m and gamma), considering both classical and Bayesian approaches. Under the Bayesian approach, we provide in a simple way necessary and sufficient conditions to check whether or not objective priors lead proper posterior distributions for the Nakagami, gamma, and GG distributions. As a result, one can easily check if the obtained posterior is proper or improper directly looking at the behavior of the improper prior. These theorems are applied to different objective priors such as Jeffreys's rule, Jeffreys prior, maximal data information prior and reference priors. Simulation studies were conducted to investigate the performance of the Bayes estimators. Moreover, maximum a posteriori (MAP) estimators for the Nakagami and gamma distribution that have simple closed-form expressions are proposed Numerical results demonstrate that the MAP estimators outperform the existing estimation procedures and produce almost unbiased estimates for the fading parameter even for a small sample size. Finally, a new lifetime distribution that is expressed as a two-component mixture of the GG distribution is presented.