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

Abstract

In their fundamental paper on cubic variance functions (VFs), Letac and Mora (The Annals of Statistics, 1990) presented a systematic, rigorous and comprehensive study of natural exponential families (NEFs) on the real line, their characterization through their VFs and mean value parameterization. They derived a construction of VFs associated with NEFs of counting distributions on the set of nonnegative integers allowing to find the corresponding generating measures. As EDMs are based on NEFs, we introduce in this paper two new classes of EDMs based on their results. For these classes, we derive their mean value parameterization and their associated generating measures. We also prove that they have some desirable properties, such as overdispersion and zero inflation in ascending order, making them as competitive statistical models for those in use in both, statistical and actuarial modeling. To our best knowledge, our classes of counting distributions, 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.

Exponential dispersion models for overdispersed zero-inflated count data

“…It is easy to see that both two classes show overdispersion, V p ðmÞ > m, and that the dispersion increases with the power r in the expressions ( 8)-( 9) of the variance functions. Moreover, in Bar-Lev and Ridder (2021) we have shown that both classes satisfy also an increasing zero-inflation property. This means, denoting f ðm, pÞ ð0; rÞ for the probability mass in zero given mean m, dispersion parameter p and power r in the variance function, then f ðm, pÞ ð0; rÞ < f ðm, pÞ ð0; r þ 1Þ, r ¼ 1, 2, ::::…”

“…Here we present the two models (or families) of distributions that were considered in Bar-Lev and Ridder (2021) as a means of a general framework for zero-inflated, overdispersed probability distributions. The models are defined through their variance function classes of the mean parameterization.…”

“…Indeed, this is our starting point: we consider a number of sets of different count data, each from another application, and our goal is to fit suitable discrete distributions. In Bar-Lev and Ridder (2021) we have developed a framework of exponential dispersion models for count data, which resulted in two classes of parametric families of discrete distributions presented in terms of their variance functions. We proved that these two classes have some desirable properties.…”

Exponential Dispersion Models for Overdispersed Zero-Inflated Count Data

“…Here we present the three models (or families) of distributions that we introduced and analysed in Bar-Lev and Ridder (2020a). The models are defined through their variance function classes of the mean parameterization.…”

“…Indeed, this is our starting point: we consider a number of sets of different count data, each from another application, and our goal is to fit suitable discrete distributions. In Bar-Lev and Ridder (2020a) we have developed a framework of exponential dispersion models for count data, which resulted in three classes of parametric families of discrete distributions presented in terms of their variance functions. We proved that these three classes have some desirable properties.…”