Abstract
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.
Highlights
Count data occur in many fields such as public health, medicine, business, physics, and epidemiology among others
The COM-Poisson distribution is well known for its flexibility and robustness in the modelling of dispersed count data
We have shown and demonstrated how the COM-Poisson distribution can be parameterized through the mean
Summary
Count data occur in many fields such as public health, medicine, business, physics, and epidemiology among others. We present a re-parameterization of the COM-Poisson distribution via the central location (mean) by solving nonlinear system of equations using the derivative-free spectral algorithm (DF-SANE). The motivation behind this approach is to provide a fast technique for fitting COM-Poisson models with easy interpretation as other competing models such as the Poisson, Generalized Poisson and the Negative Binomial distributions while retaining the attractive properties of the standard COM-Poisson distribution. Given a random sample y1, y2, ... , yn, it can be shown that the ML estimator of μ is the sample mean
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