Fitting Com-Poisson Mixtures to Bimodal Count Data

Bi-modal truncated count distributions are frequently observed in aggregate surveys and ratings when respondents are mixed in their opinion. They also arise in censored count data, where the highest category might create an additional mode. The Poisson distribution is the most common distribution for fitting count data and can be modified to achieve mixtures of truncated Poisson distributions. However, it is suitable only for modeling equi-dispersed distributions and is limited in its ability to capture bi-modality. The Conway-Maxwell-Poisson (CMP) distribution is a two-parameter generalization of the Poisson distribution that allows for overand under-dispersion. While the CMP is much more flexible, it still cannot capture bi-modality. In this work, we propose a mixture of CMPs for capturing a wide range of truncated count data, which can exhibit unimodal behavior (with equi-, underor over-dispersion) as well as bimodal behaviour. We present methods for estimating the parameters of a mixture of two CMP distributions using an EM approach. Our approach introduces a special two-step optimization within the M-step to estimate multiple parameters. The methods are illustrated using simulated and real data.