Open AccessParameterization of the COM-Poisson Distribution through the Mean Using Spectral Algorithms for Solving Nonlinear Equations
Author(s) -
Isaac Adeola Adeniyi,
Dolapo Shobanke,
Helen Olaronke Edogbanya
Publication year - 2019
Publication title -
pakistan journal of statistics and operation research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.354
H-Index - 15
eISSN - 2220-5810
pISSN - 1816-2711
DOI - 10.18187/pjsor.v15i3.2668
Subject(s) - poisson distribution , mathematics , compound poisson distribution , quasi likelihood , poisson binomial distribution , count data , negative binomial distribution , zero inflated model , compound poisson process , nonlinear system , distribution (mathematics) , poisson regression , statistics , mathematical analysis , beta binomial distribution , population , physics , demography , quantum mechanics , sociology , poisson process
The Poisson regression is popularly used to model count data. However, real data often do not satisfy the assumption of equality of the mean and variance which is an important property of the Poisson distribution. The Poisson – Gamma (Negative binomial) distribution and the recent Conway-Maxwell-Poisson (COM-Poisson) distributions are some of the proposed models for over- and under-dispersion respectively. Nevertheless, the parameterization of the COM-Poisson distribution still remains a major challenge in practice as the location parameter of the original COM-Poisson distribution rarely represents the mean of the distribution. As a result, this paper proposes a new parameterization of the COM-Poisson distribution via the central location (mean) so that more easily-interpretable models and results can be obtained. The parameterization involves solving nonlinear equations which do not have analytical solutions. The nonlinear equations are solved using the efficient and fast derivative free spectral algorithm. Implementation of the parameterization in R (R Core Team, 2018) is used to present useful numerical results concerning the relationship between the mean of the COM-Poisson distribution and the location parameter in the original COM-Poisson parameterization. The proposed technique is further used to fit COM-Poisson probability models to real life datasets. It was found that obtaining estimates via this parameterization makes the estimation easier and faster compared to directly maximizing the likelihood function of the standard COM-Poisson distribution.