  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.
