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

. Even if the most powerful tests are nonrandomized, this may be false. Suppose X takes values 1, 2, and 3 with probabilities 0.85, 0.1, and 0.05 under H and probabilities 0.7, 0.2, and 0.1 under K respectively.

(i) At any level < 0.15, the MP test is not unique.

(ii) At α = 0.05 and α = 0.1, there exist unique nonrandomized MP tests and they are not nested.

(iii) At these levels there exist MP tests φ and φ that are nested in the sense that

φ(x) ≤ φ

(x) for all x. [This example appears as Example 10.16 in Romano and Siegel (1986).]