  Bimodal truncated count distributions are frequently observed in aggregate
survey data and in user ratings when respondents are mixed in their opinion.
They also arise in censored count data, where the highest category might create
an additional mode. Modeling bimodal behavior in discrete data is useful for
various purposes, from comparing shapes of different samples (or survey
questions) to predicting future ratings by new raters. 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 bimodality. The Conway-Maxwell-Poisson (CMP) distribution is a
two-parameter generalization of the Poisson distribution that allows for over-
and under-dispersion. In this work, we propose a mixture of CMPs for capturing
a wide range of truncated discrete data, which can exhibit unimodal and bimodal
behavior. 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. We
examine computational and theoretical issues. The methods are illustrated for
modeling ordered rating data as well as truncated count data, using simulated
and real examples.
