Consider the situation of Example 6.3.1 with n = 1, and suppose that f is strictly increasing on (0, 1).

(i) The likelihood ratio test rejects if X < α/2 or X > 1 − α/2.

(ii) The MP invariant test agrees with the likelihood ratio test when f is convex.

(iii) When f is concave, the MP invariant test rejects when 1

2 − α

2

2 , and the likelihood ratio test is the least powerful invariant test against both alternatives and has power < α.