Goodness-of-Fit Tests for COM-Poisson Distribution Using Stein's Characterization

Abstract

A crucial part of data analysis is to assess the adequacy of the fit of the probability models used, and such assessment is generally made using the goodness-of-fit tests. In this context, we propose goodness-of-fit tests for the two-parameter generalization of the Poisson distribution known as the Conway-Maxwell Poisson (COM-Poisson) distribution. The ability of COM-Poisson distribution to handle a wide range of dispersion makes it a suitable candidate for modelling discrete data. The usual goodness-of-fit tests like chi-square, Cramér–von Mises and Anderson-Darling tests, when applied to COM-Poisson distribution, are computationally complex due to the presence of the normalizing constant in the probability mass function. In this article, we overcome this complexity by representing the probability mass function of the COM-Poisson distribution using Stein's characterization and propose modified test statistics. The performance of the modified test statistics is assessed in terms of the empirical level and percentage of rejection through simulation study. Applicability of the modified tests is demonstrated through real data sets.