New mixture distributions for modelling count data

A class of new 1-parameter underdispersed distributions is introduced. Mixed with Poisson distributions; they generate 2- and 3-parameter discrete distributions that generalize the Poisson distribution and can be both under and over-dispersed. Probabilities are easy to compute and moments and random number generation are tractable. The distributions are described, and they are fitted to some underdispersed and overdispersed datasets. We show how inference for the effect of covariates sharpens on moving from the Poisson model. The fits compare favourably to two benchmarks, the COM Poisson distribution and the weighted Poisson distribution.