Consistency and asymptotic normality of the maximum likelihood estimator in a zero-inflated generalized Poisson regression

Poisson regression models for count variables have been utilized in
many applications. However, in many problems overdispersion and
zero-inflation occur. We study in this paper regression models
based on the generalized Poisson distribution (Consul (1989)).
These regression models which have been used for about 15 years do
not belong to the class of generalized linear models considered by
(McCullagh and Nelder (1989)) for which an established asymptotic
theory is available. Therefore we prove consistency and asymptotic
normality of a solution to the maximum likelihood equations for
zero-inflated generalized Poisson regression models. Further the
accuracy of the asymptotic normality approximation is investigated
through a simulation study. This allows to construct asymptotic
confidence intervals and likelihood ratio tests.