Overdispersion: Models and estimation

Overdispersion models for discrete data are considered and placed in a general framework. A distinction is made between completely specified models and those with only a mean-variance specification. Different formulations for the overdispersion mechanism can lead to different variance functions which can be placed within a general family. In addition, many different estimation methods have been proposed, including maximum likelihood, moment methods, extended quasi-likelihood, pseudo-likelihood and non-parametric maximum likelihood. We explore the relationships between these methods and examine their application to a number of standard examples for count and proportion data. A simple graphical method using half-normal plots is used to examine different overdispersion models.