On asymptotically optimal tests of composite hypotheses

Suppose we are given sample observations from a distribution f(X|θ1, …, θk), and it is desired to test the null hypothesis θ1 = 0 against the alternative hypothesis that θ1 ≠ 0, the values of θ2, …, θk being unknown. Various asymptotically optimal procedures for doing this are considered. It is shown that the C(α) tests used by Neyman are asymptotically equivalent to the use of the likelihood ratio test and to tests using the maximum likelihood estimators. The necessary conditions for C(α) tests to be asymptotically optimal are re-examined and the way in which they can be applied to randomized and nonrandomized experiments explained and illustrated. The generalization of these tests to cases where the alternative hypothesis involves changes in more than one parameter is also mentioned.