Empirical Bayes analysis for a hierarchical Poisson generalized linear model

We describe a hierarchical Poisson regression model for count data with possible extra-Poisson variation. This model depicts the effects of explanatory variables through a generalized linear model embedded at the prior level of the hierarchy. Model parameters are estimated in a parametric empirical Bayes framework under various estimation schemes. Properties of consistency and asymptotic normality for the hyperparameter estimators are established under the assumption that the number of observations at each treatment (or treatment combination) are large while the number of treatment levels remains fixed. These asymptotic properties form the basis for the large sample inferences on the hyperparameters. An example illustrates use of the methodology in practice.