Realization of closed, convex, and symmetric subsets of the unit square as regions of risk for testing simple hypotheses

Abstract

It is well-known that the region of risk for testing simple hypotheses is some closed, convex, and (1/2, 1/2)-symmetric subset of the unit square, which contains the points (0, 0) and (1, 1). It is shown that for any such subsetR of the unit square and any atomless probability measureP on some σ-algebra there exists some probability measureQ on the same σ-algebra such thatR is the corresponding region of risk for testingP againstQ. This generalizes a result of [4] and is as a first step derived here for the special case, whereP is equal to the uniform distribution on the unit interval. The corresponding distributionQ is given explicitly in this case and the general case is treated by some well-known measure-isomorphism. This method of proof shows thatQ might be chosen to be of typeQ=λQ 1+(1−λ)Q 2 for some λ satisfying 0≤λ≤1, whereQ 1 is a probability measure, which is absolutely continuous with respect toP andQ 2 is a one-point mass.