Proof of a necessary and sufficient condition for admissibility in discrete multivariate problems

The proof of Farrell (1968. Ann. Math. Statist. 26 518-522) is adapted to the special problems presented by discrete problems. Continuity of the risk functions is verified, sequential subcompactness is verified, and a necessary and sufficient condition for admissibility proven. In the discrete problems considered one obtains pointwise convergence of the sequence of Bayes estimators to the admissible estimator. This last property is crucial to further development of the decision theory given in Brown and Farrell (1985. Ann. Math. Statist. 13 706-726).