Impartial Decision Rules and Sufficient Statistics

Abstract

A class of decision problems concerning $k$ populations was considered in [1] and it was shown that a particular decision rule is the uniformly best `impartial' decision rule for many problems of this class. The present paper provides certain improvements of this result. The authors define impartiality in terms of permutations of the $k$ samples rather than in terms of the $k$ ordered values of an arbitrarily chosen real-valued statistic as in the earlier paper. They point out that (under conditions which are satisfied in the standard cases of $k$ independent samples of equal size) if the same function is a sufficient statistic for each of the $k$ samples then the conditional expectation of an impartial decision rule given the $k$ sufficient statistics is also an impartial decision rule. A characterization of impartial decision rules is given which relates the present definition of impartiality with the one adopted in [1]. These results, together with Theorem 1 of [1], yield the desired improvements. The argument indicated here is illustrated by application to a special case.