Mittag - Leffler function distribution - A new generalization of hyper-Poisson distribution

TL;DR

This paper introduces the Mittag-Leffler function distribution (MLFD), a flexible new generalization of the hyper-Poisson distribution, capable of modeling various dispersion types and distribution shapes for count data.

Contribution

The paper proposes MLFD, a novel distribution based on Mittag-Leffler functions, extending hyper-Poisson and related distributions with comprehensive properties and empirical applicability.

Findings

MLFD can model under, equi, and over dispersed data

It exhibits diverse shapes including unimodal and non-increasing

MLFD compares favorably with hyper-Poisson in empirical data fitting

Abstract

In this paper a new generalization of the hyper-Poisson distribution is proposed using the Mittag-Leffler function. The hyper-Poisson, displaced Poisson, Poisson and geometric distributions among others are seen as particular cases. This Mittag-Leffler function distribution (MLFD) belongs to the generalized hypergeometric and generalized power series families and also arises as weighted Poison distributions. MLFD is a flexible distribution with varying shapes like non-increasing with unique mode at zero, unimodal with one / two non-zero modes. It can be under, equi or over dispersed. Various distributional properties like recurrence relation for pmf, cumulative distribution function, generating functions, formulae for different type of moments, their recurrence relations, index of dispersion, its classification, log-concavity, reliability properties like survival, increasing failure rate, unimodality, and stochastic ordering with respect to hyper-Poisson distribution have been discussed. The distribution has been found to fare well when compared with the hyper-Poisson distributions in its suitability in empirical modeling of differently dispersed count data. It is therefore expected that proposed MLFD with its interesting features and flexibility, will be a useful addition as a model for count data.