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 $ x = ( x _ {1} \dots x _ {n} ) $ of a random vector $ X = ( X _ {1} \dots X _ {n} ) $, the hypothesis $ H _ {0} $ to be tested is whether or not the unknown probability density of $ X $ is symmetric with respect to permutations of the arguments, that is, whether

$$ p ( x _ {1} \dots x _ {n} ) = \ p ( x _ {r _ {1} } \dots x _ {r _ {n} } ) , $$

where $ ( r _ {1} \dots r _ {n} ) $ is an arbitrary permutation of $ ( 1 \dots n ) $. Let $ X ^ {( \cdot ) } $ and $ R $ be the vector of order statistics (cf. Order statistic) and the rank vector, respectively, constructed from $ X $, and let a statistic $ \Psi = \Psi ( X ^ {( \cdot ) } , R ) $ with values in $ [ 0, 1 ] $ be such that for some $ \alpha \in ( 0 , 1 ) $,

$$ {\mathsf E} \{ \Psi ( X ^ {( \cdot ) } , R ) \mid X ^ {( \cdot ) } \} = \alpha $$

almost-everywhere. Then the statistical test with critical function $ \phi $ connected with $ \Psi $ by the relation $ \phi ( X) = \Psi ( X ^ {( \cdot ) } , R ) $ is called a randomization test. Since $ X ^ {( \cdot ) } $ 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)