Exponential Dispersion Models

SUMMARY We study general properties of the class of exponential dispersion models, which is the multivariate generalization of the error distribution of Nelder and Wedderburn’s (1972) generalized linear models. Since any given moment generating function generates an exponential dispersion model, there exists a multitude of exponential dispersion models, and some new examples are introduced. General results on convolution and asymptotic normality of exponential dispersion models are presented. Asymptotic theory is discussed, including a new small-dispersion asymptotic framework, which extends the domain of application of large-sample theory. Procedures for constructing new exponential dispersion models for correlated data are introduced, including models for longitudinal data and variance components. The results of the paper unify and generalize standard results for distributions such as the Poisson, the binomial, the negative binomial, the normal, the gamma, and the inverse Gaussian distributions.