View source for Most-powerful test

A statistical test that has the greatest power of all tests with the same [[Significance level|significance level]]. Suppose that the results of observations are to be used to test a hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m0650201.png" /> against a simple alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m0650202.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m0650203.png" /> be the (given) admissible probability of an error of the first kind — the error arising from the rejection of the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m0650204.png" /> being tested, according to a statistical test devised to test <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m0650205.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m0650206.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m0650207.png" /> is actually true. In the theory of statistical hypotheses testing, the best test of all those intended for testing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m0650208.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m0650209.png" /> and offering the same probability of an error of the first kind, or, equivalently, having the same significance level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m06502010.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m06502011.png" /> being tested when the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m06502012.png" /> is true. This best test is called the most-powerful test of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m06502013.png" /> among all tests of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m06502014.png" /> intended for testing a simple hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m06502015.png" /> against a simple alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m06502016.png" />. Since the power of a statistical test is one minus the probability of an error of the second kind, the error arising from accepting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m06502017.png" /> 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|Neyman–Pearson lemma]], according to which the [[Likelihood-ratio test|likelihood-ratio test]] is a most-powerful test.

If the competing hypotheses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m06502018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065020/m06502019.png" /> are compound, the problem of devising most-powerful tests is formulated in terms of a [[Uniformly most-powerful test|uniformly most-powerful test]], if such a test exists.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR></table>