Characterization of Risk Sets for Simple versus Simple Hypothesis Testing

Consider testing a simple hypothesis H 0: F = F 0 against a simple alternative H 1: F = F 1 when a random variable X having distribution function F is observed.The risk region is defined to be the set of ordered pairs of error probabilities (E 0[.(X)], E 1[l - .(X)]) as . ranges throughout all tests.This set plays an important role in the Neyman-Pearson lemma, which characterizes the class of most powerful tests of H 0 versus H 1.In a typical introductory course in hypothesis testing, the risk region is shown to be a closed convex subset of the unit square [0, 1] x [0, 1] that is symmetric about (1/2, 1/2) and contains the points (1, 0) and (0, 1).In this article, it is shown that any set with these properties is the risk region for some pair of hypotheses.In fact, one can take F 0 to be the uniform distribution on [0, 1].This result provides an exercise that illustrates the applicability of some results about convexity and absolute continuity that would be presented in a real analysis course at the level of Royden (1968).