On the existence of maximum likelihood estimators in Poisson-gamma HGLM and negative binomial regression model

Abstract

A breakthrough is provided in the study of the existence problem for maximum likelihood estimators (MLE) in the hierarchical generalized linear model (HGLM) of Poisson-gamma type, as well as in the negative binomial regression model. Any more than the uniqueness problem associated, the existence problem of MLE for these models has not yet been studied except in the very special case of the sample. This issue is addressed here for the Poisson-gamma HGLM, and a sufficient condition is obtained to ensure the MLE existence in that case. It is also shown that this condition has the same effect in the negative binomial regression model with the index parameter considered as unknown. In the latter model, the obtained condition appears as a natural extension of the necessary and sufficient condition well known for solving the existence and uniqueness problems for the index parameter MLE in the sample case.