A counterexample. Typically, as varies the most powerful level tests for testing a hypothesis H against a simple alternative are nested in the

A counterexample. Typically, as α varies the most powerful level

α tests for testing a hypothesis H against a simple alternative are nested in the sense that the associated rejection regions, say Rα, satisfy Rα ⊂ Rα
, for any α <

α

. The following example shows that this need not be satisfied for composite H.

Let X take on the values 1, 2, 3, 4 with probabilities under distributions P0, P1, Q:

1234 P0 2 13 4

13 3

13 4

13 P1 4 13 2

13 1

13 6

13 Q 4 13 3

13 2

13 4

13 Then the most powerful test for testing the hypothesis that the distribution of X is P0 or P1 against the alternative that it is Q rejects at level α = 5 13 when X = 1 or 3, and at level α = 6 13 when X = 1 or 2.