Is the Selected Population the Best?

Abstract

Random variables X_i ~ N(theta_i, 1), i = 1, 2, ..., k, are observed. Suppose X_S is the largest observation. If the inference theta_S > max_{i not equal S} theta_i is made whenever X_S - max_{i not equal S} X_i > c, then the probability of a false inference is maximized when two theta_i are equal and the rest are -infty. Equivalently, the inference can be made whenever a two-sample two-sided test for difference of means, based on the largest two observations, would reject the hypothesis of no difference. The result also holds in the case of unknown, estimable, common variance, and in fact for location families with monotone likelihood ratio.