34. A counterexample. Typically, as a varies the most powerful level-a tests for testing a hypothesis H against a simple alternative are nested in the sense that the associated rejection regions, say Ra, satisfy Ra eRa' for any a < a' . This relation always holds when H is simple, but the following example shows that it need not be satisfied for composite H. Let X take on the values 1,2,3,4 with probabilities under distributions Po, PI' Q: 1 2 3 4 Po .1.- ..!. .a. ..!. 13 13 13 13 PI ..!. .1.- .i, 13 13 13 13 Q ..!. .l. .l. 4 13 13 13 TI Then the most powerful test for testing the hypothesis that the distribution of X is Po or PI against the alternative that it is Q rejects at level a = fJ when X = 1 or 3, and at level a = fJ when X = 1 or 2.