On the existence of tests uniformly more powerful than the likelihood ratio test

Liu, Wei (1995) On the existence of tests uniformly more powerful than the likelihood ratio test. Journal of Statistical Planning and Inference, 44 (1), 19-35. (doi:10.1016/0378-3758(94)00036-U).

Record type: Article

Abstract

For hypotheses concerning linear inequalities and k normal means, Berger (J. Amer. Statist. Assoc.84 (1989) 192–199) showed that the likelihood ratio test (LRT) is not very powerful, and uniformly more powerful tests which reject in some extra regions can be constructed under some conditions. In this article, we study whether uniformly more powerful tests of the form considered by Berger (1989) exist in some situations that are not covered by the results of Berger (1989). First, when the covariance matrix has a special structure which usually arises from the problem of comparing several treatments with a control, we show that there exists no such uniformly more powerful test for k = 2, and there always exists such a uniformly more powerful test for k ? 3. Secondly, when the common variance of the k independent normal populations is unknown, we again show the nonexistence of such a uniformly more powerful test for k = 2, and the existence of such a uniformly more powerful test for k ? 3. The method of proof can be applied to other situations, though it does not tell how to construct such a uniformly more powerful test.

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