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''permutation 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.
 
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
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Given a realization $  x = ( x _ {1} \dots x _ {n} ) $
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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
  
<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>
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$$
 +
p ( x _ {1} \dots x _ {n} )  = \
 +
p ( x _ {r _ {1}  } \dots x _ {r _ {n}  } ) ,
 +
$$
  
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" />,
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where $  ( r _ {1} \dots r _ {n} ) $
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is an arbitrary permutation of $  ( 1 \dots n ) $.  
 +
Let $  X ^ {( \cdot ) } $
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and $  R $
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be the vector of order statistics (cf. [[Order statistic|Order statistic]]) and the [[Rank vector|rank vector]], respectively, constructed from $  X $,  
 +
and let a statistic $  \Psi = \Psi ( X ^ {( \cdot ) } , R ) $
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with values in $  [ 0, 1 ] $
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be such that for some $  \alpha \in ( 0 , 1 ) $,
  
<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>
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$$
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{\mathsf E} \{ \Psi ( X ^ {( \cdot ) } , R ) \mid
 +
X ^ {( \cdot ) } \}
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= \alpha
 +
$$
  
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.
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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|sufficient statistic]], the family of similar tests (cf. [[Similar test|Similar test]]) coincides with the family of permutation tests.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:09, 6 June 2020


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)
How to Cite This Entry:
Randomization test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Randomization_test&oldid=48431
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article