Re-parameterization of the COM-Poisson Distribution Using Spectral Algorithms

The Poisson regression is popularly used to model count data but real life data often do not satisfy the essential assumption of equality of the mean and variance of the Poisson distribution. The Conway-Maxwell-Poisson (COM-Poisson) distributions is one of the models that have been proposed for handling cases of over-and under-dispersion. 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 nonlinear equations resulting from the re-parameterization were solved using the efficient and fast derivative free spectral algorithm. The proposed parameterization is used to present useful numerical results concerning the mean of the COM-Poisson distribution and the location parameter in the original COM-Poisson parameterization. Application of the re-parameterization is further illustrated by fitting COM-Poisson probability models to real life datasets.