Generalised Poisson Distributions for Modelling Parity

Conventional parametric count distributions, namely the Poisson and Negative-Binomial models, do not offer satisfactory descriptions of empirical distributions of completed cohort parity. One reason is that they cannot model variance-to-mean ratios below unity, that is, underdispersion, which is typical of low-fertility parity distributions. Statisticians have relatively recently revived two generalised count distributions that can model both overdispersion and underdispersion, but that have to date not attracted the attention of demographers. The objective of this note is to assess the utility of these distributions, the Conway-Maxwell-Poisson and Gamma Count models, for the modelling of parity distributions, using both simulations and maximum-likelihood fitting to empirical data from the Human Fertility Database (HFD). The results show that these generalised count distributions offer a greatly improved fit compared to customary Poisson and Negative-Binomial models in the presence of underdispersion, without loss of performance in the presence of equi- or overdispersion.