A new poisson-exponential-gamma distribution for modelling count data with applications

In this paper, a new member of the Poisson family of distributions called the Poisson-Exponential-Gamma (PEG) distribution for modelling count data is proposed by compounding the Poisson with Exponential-Gamma distribution. The first four moments about the origin and the mean of the new PEG distribution were obtained. The expressions for its coefficient of variation, skewness, kurtosis, and index of dispersion were equally derived. The parameters of the PEG distribution were estimated using the Maximum Likelihood Method. Its relative performance based on the Goodness-of-Fit (GoF) criteria was compared with those provided by seven of the existing related distributions (Poisson, Negative-Binomial, Poisson-Exponential, Poisson-Lindley, Poisson-Shanker, Poisson-Shukla, and Poisson Entropy-Based Weighted Exponential distributions) in the literature on three different published real-life count data sets. The GoF assessment of all these distributions was performed based on the values of their loglikelihoods ( $${-}2{\text{logLik}}$$ - 2 logLik ), Akaike Information Criteria, Akaike Information Criteria Corrected, and Bayesian Information Criteria. The results showed that the new PEG distribution was relatively more efficient for modelling (over-dispersed) count data than any of the seven existing distributions considered. The new PEG distribution is therefore recommended as a credible alternative for modelling count data whenever relative gain in the model’s efficiency is desired.