Abstract
Within the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD’s) to allow both overdispersion and underdispersion. Similarly to Kemp’s generalized hypergeometric probability distribution, which is based on hypergeometric functions, we analyze a class of WPD’s related to a generalization of Mittag–Leffler functions. The proposed class of distributions includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature.
Highlights
Introduction and mathematical backgroundThe negative binomial distribution is one of the most widely used discrete probability models that allow departure from the mean-equal-variance Poisson model
An example is given in Sect. 2.1.1, in which we introduce two real world data sets that do not fit a negative binomial model
The generalized fractional Poisson distribution, which we introduce lies in the same class of the Kemp’s generalized hypergeometric factorial moments distributions (GHFD) but with the hypergeometric function in (9) substituted by a generalized Mittag–Leffler function
Summary
The negative binomial distribution is one of the most widely used discrete probability models that allow departure from the mean-equal-variance Poisson model. The generalized fractional Poisson distribution (gfPd), which we introduce, lies in the same class of the Kemp’s GHFD but with the hypergeometric function in (9) substituted by a generalized Mittag–Leffler function ( known as three-parameter Mittag–Leffler function or Prabhakar function) In this case, as we have anticipated above, the model is capable of describing overdispersion and having a degree of flexibility in dealing with skewness. It is worthy to note that there exists a second family of Kemp’s distributions, still based on hypergeometric functions and still allowing both underdispersion and overdispersion This is known the Kemp’s generalized hypergeometric probability distribution (GHPD) (Kemp 1968) and it is a special case of the very general class of weighted Poisson distributions (WPD). 2, we introduce the generalized fractional Poisson distribution, discuss some properties and recover the classical fPd as a special case These models are fit to the two real-world data sets mentioned above. For δ ∈ (0, β/α) the bound (20) is always verified
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