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A [[Non-parametric test|non-parametric test]] of the homogeneity of two samples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w0979601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w0979602.png" />. The elements of the samples are assumed to be mutually independent, with continuous distribution functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w0979603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w0979604.png" />, respectively. The hypothesis to be tested is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w0979605.png" />. Wilcoxon's test is based on the [[Rank statistic|rank statistic]]
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<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/w/w097/w097960/w0979606.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w0979607.png" /> are the ranks of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w0979608.png" /> in the common series of order statistics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w0979609.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w09796010.png" />, while the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w09796011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w09796012.png" />, is defined by a given permutation
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A [[Non-parametric test|non-parametric test]] of the homogeneity of two samples  $  X _ {1} \dots X _ {n} $
 +
and  $  Y _ {1} \dots Y _ {m} $.  
 +
The elements of the samples are assumed to be mutually independent, with continuous distribution functions  $  F( x) $
 +
and $  G( x) $,  
 +
respectively. The hypothesis to be tested is  $  F( x)= G( x) $.  
 +
Wilcoxon's test is based on the [[Rank statistic|rank statistic]]
  
<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/w/w097/w097960/w09796013.png" /></td> </tr></table>
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$$ \tag{* }
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= s ( r _ {1} ) + \dots + s ( r _ {m} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w09796014.png" /> is one of the possible rearrangements of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w09796015.png" />. The permutation is chosen so that the power of Wilcoxon's test for the given alternative is highest. The statistical distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w09796016.png" /> depends only on the size of the samples and not on the chosen permutation (if the homogeneity hypothesis is true). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w09796017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w09796018.png" />, the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w09796019.png" /> has an asymptotically-normal distribution. This variant of the test was first proposed by F. Wilcoxon in 1945 for samples of equal sizes and was based on the special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097960/w09796020.png" /> (cf. [[Rank sum test|Rank sum test]]; [[Mann–Whitney test|Mann–Whitney test]]). See also [[Van der Waerden test|van der Waerden test]]; [[Rank test|Rank test]].
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where $  r _ {j} $
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are the ranks of the random variables  $  Y _ {j} $
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in the common series of order statistics of  $  X _ {i} $
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and  $  Y _ {j} $,
 +
while the function  $  s( r) $,
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$  r = 1 \dots n + m $,
 +
is defined by a given permutation
 +
 
 +
$$
 +
\left(
 +
\begin{array}{cccc}
 +
1 & 2 & \cdots & m+n \\
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s(1) & s(2) & \cdots & s(m+n)
 +
\end{array}
 +
\right)\ ,
 +
$$
 +
where  $  s( 1) \dots s( n+ m) $
 +
is one of the possible rearrangements of the numbers $  1 \dots n + m $.  
 +
The permutation is chosen so that the power of Wilcoxon's test for the given alternative is highest. The statistical distribution of $  W $
 +
depends only on the size of the samples and not on the chosen permutation (if the homogeneity hypothesis is true). If $  n \rightarrow \infty $
 +
and $  m \rightarrow \infty $,  
 +
the random variable $  W $
 +
has an asymptotically-normal distribution. This variant of the test was first proposed by F. Wilcoxon in 1945 for samples of equal sizes and was based on the special case $  s( r) \equiv r $(
 +
cf. [[Rank sum test|Rank sum test]]; [[Mann–Whitney test|Mann–Whitney test]]). See also [[Van der Waerden test|van der Waerden test]]; [[Rank test|Rank test]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Wilcoxon,  "Individual comparison by ranking methods"  ''Biometrics'' , '''1''' :  6  (1945)  pp. 80–83</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.N. Bol'shev,  N.V. Smirnov,  "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka  (1983)  (In Russian)  (Processed by L.S. Bark and E.S. Kedrova)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  F. Wilcoxon,  "Individual comparison by ranking methods"  ''Biometrics'' , '''1''' :  6  (1945)  pp. 80–83</TD></TR>
 
+
<TR><TD valign="top">[2]</TD> <TD valign="top">  L.N. Bol'shev,  N.V. Smirnov,  "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka  (1983)  (In Russian)  (Processed by L.S. Bark and E.S. Kedrova)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR>
 +
</table>

Latest revision as of 10:34, 7 February 2021


A non-parametric test of the homogeneity of two samples $ X _ {1} \dots X _ {n} $ and $ Y _ {1} \dots Y _ {m} $. The elements of the samples are assumed to be mutually independent, with continuous distribution functions $ F( x) $ and $ G( x) $, respectively. The hypothesis to be tested is $ F( x)= G( x) $. Wilcoxon's test is based on the rank statistic

$$ \tag{* } W = s ( r _ {1} ) + \dots + s ( r _ {m} ), $$

where $ r _ {j} $ are the ranks of the random variables $ Y _ {j} $ in the common series of order statistics of $ X _ {i} $ and $ Y _ {j} $, while the function $ s( r) $, $ r = 1 \dots n + m $, is defined by a given permutation

$$ \left( \begin{array}{cccc} 1 & 2 & \cdots & m+n \\ s(1) & s(2) & \cdots & s(m+n) \end{array} \right)\ , $$ where $ s( 1) \dots s( n+ m) $ is one of the possible rearrangements of the numbers $ 1 \dots n + m $. The permutation is chosen so that the power of Wilcoxon's test for the given alternative is highest. The statistical distribution of $ W $ depends only on the size of the samples and not on the chosen permutation (if the homogeneity hypothesis is true). If $ n \rightarrow \infty $ and $ m \rightarrow \infty $, the random variable $ W $ has an asymptotically-normal distribution. This variant of the test was first proposed by F. Wilcoxon in 1945 for samples of equal sizes and was based on the special case $ s( r) \equiv r $( cf. Rank sum test; Mann–Whitney test). See also van der Waerden test; Rank test.

References

[1] F. Wilcoxon, "Individual comparison by ranking methods" Biometrics , 1 : 6 (1945) pp. 80–83
[2] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)
[3] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)

Comments

References

[a1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
How to Cite This Entry:
Wilcoxon test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wilcoxon_test&oldid=16296
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article