$\Gamma$-Minimax Selection Procedures in Simultaneous Testing Problems

Abstract

Suppose we have to decide on the basis of appropriately drawn samples which of $k$ treatment populations are "good" compared to either given control values or to a control population from which an additional sample is available. The unknown parameters are assumed to vary randomly according to a prior distribution about which we only have the partial knowledge that it is contained in a given class $\Gamma$ of priors. Though we derive in both cases (under the assumption of monotone likelihood ratios) $\Gamma$-minimax procedures which by definition attain minimal supremal risk over $\Gamma$, the emphases are different: while we try to demonstrate in the "known controls case" how well known results from the theory of testing hypotheses can be utilized to solve the problem, our main purpose in the "unknown control case" is to give a new proof for a theorem which was stated but only partially proved by Randles and Hollander.