Flexible models for overdispersed and underdispersed count data

Abstract Within the framework of probability models for overdispersed count data, we propose generalized fractional Poisson distribution (gfPd), which is a natural generalization (fPd), and standard distribution. We derive some properties gfPd more specifically study moments, limiting behavior other features fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes model flexible appealing in practice. apply to real big data estimate parameters using maximum likelihood. Then, turn very general class weighted distributions (WPD’s) allow both overdispersion underdispersion. Similarly Kemp’s hypergeometric distribution, based on functions, analyze WPD’s related Mittag–Leffler functions. proposed includes well-known COM-Poisson hyper-Poisson models. characterize conditions allowing underdispersion, two special cases interest have not yet appeared literature.