Most-powerful test

A statistical test that has the greatest power of all tests with the same significance level. Suppose that the results of observations are to be used to test a hypothesis $H_0$ against a simple alternative $H_1$; let $\alpha$ be the (given) admissible probability of an error of the first kind — the error arising from the rejection of the hypothesis $H_0$ being tested, according to a statistical test devised to test $H_0$ against $H_1$, when $H_0$ is actually true. In the theory of statistical hypotheses testing, the best test of all those intended for testing $H_0$ against $H_1$ and offering the same probability of an error of the first kind, or, equivalently, having the same significance level $\alpha$, is the test that has the highest power. In other words, the best test is the one that offers the highest probability of rejecting the hypothesis $H_0$ being tested when the alternative $H_1$ is true. This best test is called the most-powerful test of level $\alpha$ among all tests of level $\alpha$ intended for testing a simple hypothesis $H_0$ against a simple alternative $H_1$. Since the power of a statistical test is one minus the probability of an error of the second kind, the error arising from accepting $H_0$ when it is in fact false, the concept of a most-powerful test is frequently formulated in terms of probabilities of errors of the first and second kinds: A most-powerful test is a test intended for testing a simple hypothesis against a simple alternative and offering the least probability of an error of the second kind among all tests with given probability of the error of the first kind. The problem of constructing most-powerful tests for simple hypotheses is solved by the Neyman–Pearson lemma, according to which the likelihood-ratio test is a most-powerful test.

If the competing hypotheses $H_0$ and $H_1$ are compound, the problem of devising most-powerful tests is formulated in terms of a uniformly most-powerful test, if such a test exists.

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[2]	J. Neyman, E. Pearson, "On the problem of the most efficient tests of statistical hypotheses" Philos. Trans. Royal Soc. London A , 231 (1933) pp. 289–337