A new over-dispersed count model based on Poisson-Geometric convolution

Abstract

A new two-parameter discrete distribution, namely the PoiG distribution, is derived by the convolution of a Poisson variate and an independently distributed geometric random variable. This distribution generalizes both the Poisson and geometric distributions and can be used for modeling over-dispersed as well as equi-dispersed count data. A number of important statistical properties of the proposed count model, such as the probability generating function, the moment generating function, the moments, the survival function and the hazard rate function have been studied in detail. Monotonic properties such as the log concavity are also studied. Stochastic ordering for the proposed model is also investigated in detail. Method of moments and the maximum-likelihood estimators of the parameters of the proposed model are found to work well in simulation experiments as well as in the real-life data analysis. Asymptotic confidence intervals and the coverage probabilities of the parameters are also studied through extensive simulation experiments. Two real life modelling applications demonstrate that the proposed distribution may prove to be useful for the practitioners for modelling over-dispersed count data compared to its closest competitors.