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

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