Randomization test

permutation test

A 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 of a random vector , the hypothesis to be tested is whether or not the unknown probability density of is symmetric with respect to permutations of the arguments, that is, whether

where is an arbitrary permutation of . Let and be the vector of order statistics (cf. Order statistic) and the rank vector, respectively, constructed from , and let a statistic with values in be such that for some ,

almost-everywhere. Then the statistical test with critical function connected with by the relation is called a randomization test. Since is a complete sufficient statistic, the family of similar tests (cf. Similar test) coincides with the family of permutation tests.

References

[1]	J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967)
[2]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)