A flexible bivariate distribution for count data expressing data dispersion

The bivariate Poisson distribution is a natural choice for modeling bivariate count data. Its constraining assumption, however, limits model flexibility in some contexts. This work considers the trivariate reduction method to construct a Bivariate Conway-Maxwell-Poisson (BCMP) distribution, which accommodates over- and under-dispersed data. The approach produces marginals that have a flexible form which includes several special case distributions for certain parameters. Moreover, this BCMP model performs well relative to other bivariate models for count data, including BCMP models based on different methods of construction. As a result, the trivariate-reduced BCMP distribution is a flexible alternative for modeling bivariate count data containing data dispersion.