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A statistical test intended for testing the hypothesis  $  H _  \star  $
according to which the probability density (cf. [[Density of a probability distribution|Density of a probability distribution]]) of an observable random vector  $  X = ( X _ {1} \dots X _ {n} ) $
belongs to the family of all  $  n $-
dimensional densities that are symmetric with respect to permutation of their arguments.

Assume that one has to test the hypothesis  $  H _  \star  $
that the probability density  $  p( x) $
of the random vector  $  X $
belongs to the family  $  \mathbf H _  \star  = \{ p( x) \} $
of all  $  n $-
dimensional densities  $  p( x) = p( x _ {1} \dots x _ {n} ) $
that are symmetric with respect to permutation of the arguments  $  x _ {1} \dots x _ {n} $, 
from a realization of the random vector  $  X = ( X _ {1} \dots X _ {n)} $
that takes values  $  x = ( x _ {1} \dots x _ {n} ) $
in  $  n $-
dimensional Euclidean space  $  \mathbf R  ^ {n} $. 
Then

$$ 
p( x)  \in  \mathbf H _  \star   \iff  p( x _ {1} \dots x _ {n} )  =  p( x _ {r _ {1}  } \dots x _ {r _ {n}  } ),
$$

where  $  r = ( r _ {1} \dots r _ {n} ) $
is any vector from the space  $  \mathfrak R $
of all permutations  $  ( r _ {1} \dots r _ {n} ) $
of the vector  $  ( 1 \dots n) $. 
The space  $  \mathfrak R $
is the set of all realizations of the vector of ranks  $  R = ( R _ {1} \dots R _ {n} ) $
naturally arising in constructing the order statistic vector  $  X  ^ {(.)} $
that takes values  $  x  ^ {(.)} $
in the set  $  \mathfrak X  ^ {(.)} \subset  \mathbf R  ^ {n} $. 
If  $  H _  \star  $
is true, then the statistics  $  X  ^ {(.)} $
and  $  R $
are stochastically independent, and

$$ \tag{* }
{\mathsf P} \{ R = r \}  =  
\frac{1}{n!}
 ,\  r \in \mathfrak R ,
$$

and the probability density for  $  X  ^ {(.)} $
is  $  n!p( x  ^ {(.)} ) $, 
$  x  ^ {(.)} \in \mathfrak X  ^ {(.)} $.

Property (*) of the uniform distribution for  $  R $
if  $  H _  \star  $
is true forms the basis of constructing the permutation test.

If  $  \Psi ( x  ^ {(.)} , r) $
is a function defined on  $  \mathfrak X  ^ {(.)} \times \mathfrak R $
in such a way that  $  0 \leq  \Psi \leq  1 $
and such that for any  $  r \in \mathfrak R $
it is measurable with respect to the Borel  $  \sigma $-
algebra of  $  \mathfrak X  ^ {(.)} $, 
and if also for some  $  \alpha \in ( 0, 1) $,

$$ 

\frac{1}{n!}
 \sum _ {r \in \mathfrak R } \Psi ( x  ^ {(.)} , r)  =  \alpha
$$

almost-everywhere, then the statistical test for testing  $  H _  \star  $
with critical function

$$ 
\phi ( x)  =  \phi ( x _ {1} \dots x _ {n} )  =  \Psi ( x  ^ {(.)} , r)
$$

is called the permutation test. If the permutation test is not randomized,  $  \alpha $
should be taken a multiple of  $  1/n! $.

The most-powerful test for testing  $  H _  \star  $
against a simple alternative  $  q( x) $
can be found in the family of permutation tests, where  $  q( x) $
is any  $  n $-
dimensional density not belonging to  $  \mathbf H _  \star  $.

The family of permutation tests and the family of tests that are invariant under a change in the shift and scale parameters play significant roles in constructing rank tests (cf. [[Rank test|Rank test]]). Finally, in the literature on mathematical statistics, one frequently finds the term permutation test replaced by  "randomization test" .

See [[Order statistic|Order statistic]]; [[Invariant test|Invariant test]]; [[Critical function|Critical function]].

====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>