Proper Bayes minimax estimation of parameters of Poisson distributions in the presence of unbalanced sample sizes

In this paper, we consider the problem of simultaneously estimating parameters of independent Poisson distributions in the presence of possibly unbalanced sample sizes under weighted standardized squared error loss. A class of heterogeneous Bayesian shrinkage estimators that utilize the unbalanced nature of sample sizes is proposed. To provide a theoretical justification, we first derive a necessary and sufficient condition for an estimator in the class to be proper Bayes and hence admissible and then obtain sufficient conditions for minimaxity that are compatible with the admissibility condition. Heterogeneous and homogeneous shrinkage estimators are compared by simulation. Several estimation methods are applied to data relating to the standardized mortality ratio.