Complete Class Theorems for Estimation of Multivariate Poisson Means and Related Problems

Basic decision theory for discrete random variables of the multivariate geometric (power series) type is developed. Some properties of Bayes estimators that carry over in the limit to admissible estimators are obtained. A stepwise generalized Bayes representation of admissible estimators is developed with estimation of the mean of a multivariate Poisson random variable in mind. The development carries over to estimation of the mean of a multivariate negative Binomial random variable. Due to the natural boundary of the parameter space there is an interesting pathology illustrated to some extent by the examples given. Examples include one to show that admissible estimators with somewhere infinite risk do exist in two or more dimensions.