Limited deviation estimators of the Poisson parameter

Estimators for the Poisson parameter are proposed which perform well with respect to both a weighted, i.e. Bayes, and unweighted risk criterion. The estimators follow the Bayes rule (with respect to a conjugate gamma prior) as closely as possible subject to a restraint imposed on the allowable deviation from the minimax estimate. The resulting class of rules maintains good performance with respect to the Bayes criterion while at the same time possessing bounded risk functions. The excess Bayes risk incurred is compared to a lower bound on the optimal restricted Bayes risk.