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In their fundamental paper on cubic variance functions, Letac and Mora (TheAnnals of Statistics,1990) presented a systematic, rigorous and comprehensivestudy of natural exponential families on the real line, their characterizationthrough their variance functions and mean value parameterization. Theypresented a section that for some reason has been left unnoticed. This sectiondeals with the construction of variance functions associated with naturalexponential families of counting distributions on the set of nonnegativeintegers and allows to find the corresponding generating measures. Asexponential dispersion models are based on natural exponential families, weintroduce in this paper two new classes of exponential dispersion models basedon their results. For these classes, which are associated with simple variancefunctions, we derive their mean value parameterization and their associatedgenerating measures. We also prove that they have some desirable properties.Both classes are shown to be overdispersed and zero-inflated in ascendingorder, making them as competitive statistical models for those in use in both,statistical and actuarial modeling. To our best knowledge, the classes ofcounting distributions we present in this paper, have not been introduced ordiscussed before in the literature. To show that our classes can serve ascompetitive statistical models for those in use (e.g., Poisson, Negativebinomial), we include a numerical example of real data. In this example, wecompare the performance of our classes with relevant competitive models.

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