Parameterizations for Natural Exponential Families with Quadratic Variance Functions

Abstract

Abstract Parameterizations for natural exponential families (NEF's) with quadratic variance functions (QVF's) are compared according to the nearness to normality of the likelihood and posterior distribution. Nonnormality of the likelihood (posterior) is measured using two criteria. The first is the magnitude of a standardized third derivative of the log-likelihood (logposterior density); the second is a comparison of the probability of particular tail regions under the normalized likelihood (posterior distribution) and under the corresponding normal approximation. A relationship is given that links these two criteria. Sample sizes are recommended for adequate normality in the likelihood for various parameterizations of the NEF-QVF models, and these results are extended to Bayesian models with a conjugate prior.