Uso das distribuições Poisson, Poisson-Gama, Poisson-Inversa Gaussiana e Poisson-Lindley Generalizada para dados de contagem

Abstract

The Generalized Linear Models (GLM) is an alternative to Normal Models when the response variable does not come from a normal distribution. When the parametric space of the the variable of interest assumes discrete and non-negative values, we study the Count Models. The common approach when analyzing these data is through a Poisson distribution. However, one of the precautions to be taken when analyzing discrete data is overdispersion. This term is used, in the area of Statistics, when the average of the variance of the response variable exceeds the nominal variance predicted 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 overdispersion. 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 associates 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, 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. Also addressed were some types of residues in the MLG structure (Pearson, deviance component, randomized quantile), adapted for the composite distributions used in the monograph. The performance of each approach was performed using real data sets described in the literature.