View source for Maximin criterion

''maximin test''

A statistical test for testing a compound hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m0630301.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m0630302.png" /> against the compound alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m0630303.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m0630304.png" />, whose minimal power (cf. [[Power of a statistical test|Power of a statistical test]]) is maximal in the class of all statistical tests intended for testing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m0630305.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m0630306.png" /> and having the same size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m0630307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m0630308.png" />. In statistically testing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m0630309.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m06303010.png" /> a maximin [[Invariant test|invariant test]] exists if the problem itself is invariant relative to some group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m06303011.png" />, and there is a [[Uniformly most-powerful test|uniformly most-powerful test]] in the class of corresponding invariant tests (cf. [[Hunt–Stein theorem|Hunt–Stein theorem]]). A maximin test exists, in general, if the family of probability distributions determined by the null and competing hypotheses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m06303012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m06303013.png" /> is absolutely continuous relative to some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063030/m06303014.png" />-finite measure.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Hájek,   Z. Sidák,   "Theory of rank tests" , Acad. Press  (1967)</TD></TR></table>