A new three-parameter discrete distribution to model over-dispersed count data

Abstract

A novel discrete distribution with three parameters, referred to as the PoiNB distribution, is formulated through the convolution of a Poisson variable and an independently distributed negative binomial random variable. This distribution generalizes some well known count distributions and can be used for modelling over-dispersed as well as equi-dispersed count data. Numerous essential statistical properties of this proposed count model are thoroughly examined. Characterizations of this distribution in terms of conditional expectation and reverse hazard rate function are studied in detail. The estimation of the unknown parameters of this proposed distribution is carried out using the maximum likelihood estimation approach. Additionally, we introduce a count regression model based on the PoiNB distribution through the generalized linear model approach. Through two real-life modelling applications, it is demonstrated that the suggested distribution may offer practical utility for practitioners in modelling over-dispersed count data.