Use of Poisson, Poisson-Gamma, Poisson-Inverse Gaussian, and Generalized Poisson-Lindley distributions for count data

Abstract

The usual analysis for discrete data is through a Poisson, Binomial or Negative Binomial distribution, via Generalized Linear Models (GLM). However, one of the precautions to be taken when analyzing discrete data is overdispersion. This term is used when the presence of variation in the data exceeds the nominal variance stipulated by the proposed model. Thus, the use of a model based on the Poisson distribution, which assumes equidispersion, would be unfounded in the presence of overdis persion. An alternative to this problem is to use mixed distributions, through models in two stages, or hierarchical, as a way to accommodate this overdispersion. The methodology of the two-stage models as sociates a distribution to the response conditioned to its average and, later, a distribution to the average parameter, so that, unconditionally, there is a compound distribution for the response variable. In this work, the classical Poisson distribution is used for counting data, and the Gama, Inverse Gaussian and  Generalized Lindley distributions for the Poisson mean parameter, thus generating the Poisson-Gama,
35 Poisson-Inverse Gaussian and Generalized Poisson-Lindley. So, the main objective of this work is to present these hierarchical models, or models in two stages, that allow the modeling of count data with overdispersion. Futhermore, some types of residues of the structure of the MLGs were also approached, adapted for the composite distributions.