View source for Randomization test

''permutation test''

A [[Statistical test|statistical test]] for the hypothesis that the probability density of the random vector under observation is symmetric with respect to permutations of its arguments.

Given a realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r0774001.png" /> of a random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r0774002.png" />, the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r0774003.png" /> to be tested is whether or not the unknown probability density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r0774004.png" /> is symmetric with respect to permutations of the arguments, that is, whether

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r0774005.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r0774006.png" /> is an arbitrary permutation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r0774007.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r0774008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r0774009.png" /> be the vector of order statistics (cf. [[Order statistic|Order statistic]]) and the [[Rank vector|rank vector]], respectively, constructed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r07740010.png" />, and let a statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r07740011.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r07740012.png" /> be such that for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r07740013.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r07740014.png" /></td> </tr></table>

almost-everywhere. Then the statistical test with critical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r07740015.png" /> connected with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r07740016.png" /> by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r07740017.png" /> is called a randomization test. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077400/r07740018.png" /> is a complete [[Sufficient statistic|sufficient statistic]], the family of similar tests (cf. [[Similar test|Similar test]]) coincides with the family of permutation tests.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hájek,   Z. Sidák,   "Theory of rank tests" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR></table>