Mean and Variance for Count Regression Models Based on Reparameterized Distributions

We introduce a new regression model for count data where the response variable is mainly in the class of inflated-parameter generalized power series (IGPS) distributions, which take automatically into account both dispersion and zero inflation phenomena. An original parameterization of these distributions is used, which is indexed by the mean and variance parameters, and not generally connected between them. An advantage of our approach is the straightforward interpretation of the regression coefficients in terms of the mean and variance comparing, for instance, to the popular generalized linear models. This attractive methodology is so simple and useful for many models. Some new mathematical and practical properties of the IGPS distributions are studied, including the quantile function, dispersion and zero-inflation indexes. Three basical IGPS models such for geometric, Bernoulli and Poisson are investigated in details. For the corresponding count regression models, the method of maximum likelihood is used for estimating the model parameters. Simulation studies are conducted to evaluate its finite sample performance. Finally, we highlight the ability of some reparameterized IGPS regression models to deal with count data which are overdispersed and zero-inflated; and then, comparing with usual models like zero inflated Poisson and negative binomial which are also reparameterized in terms of mean and variance.