New exponential dispersion models for count data -- the ABM and LM classes

In their fundamental paper on cubic variance functions, Letac and Mora (The Annals of Statistics,1990) presented a systematic, rigorous and comprehensive study of natural exponential families on the real line, their characterization through their variance functions and mean value parameterization. They presented a section that for some reason has been left unnoticed. This section deals with the construction of variance functions associated with natural exponential families of counting distributions on the set of nonnegative integers and allows to find the corresponding generating measures. As exponential dispersion models are based on natural exponential families, we introduce in this paper two new classes of exponential dispersion models based on their results. For these classes, which are associated with simple variance functions, we derive their mean value parameterization and their associated generating measures. We also prove that they have some desirable properties. Both classes are shown to be overdispersed and zero-inflated in ascending order, making them as competitive statistical models for those in use in both, statistical and actuarial modeling. To our best knowledge, the classes of counting distributions we present in this paper, have not been introduced or discussed before in the literature. To show that our classes can serve as competitive statistical models for those in use (e.g., Poisson, Negative binomial), we include a numerical example of real data. In this example, we compare the performance of our classes with relevant competitive models.