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

It is well-known that the region of risk for testing simple hypotheses is some closed, convex, and -symmetric subset of the unit square, which contains the points (0, 0) and (1, 1). It is shown that for any such subset R of the unit square and any atomless probability measure P on some -algebra there exists some probability measure Q on the same -algebra such that R is the corresponding region of risk for testing P against Q. This generalizes an earlier result and is as a first step derived here for the special case, where P is equal to the uniform distribution on the unit interval. The corresponding distribution Q is given explicitly in this case and the general case is treated by some well-known measure-isomorphism. This method of proof shows that Q might be chosen to be of type for some satisfying , where is a probability measure, which is absolutely continuous with respect to P and is a one-point mass.