View source for Wilcoxon test

<!--
w0979601.png
$#A+1 = 20 n = 0
$#C+1 = 20 : ~/encyclopedia/old_files/data/W097/W.0907960 Wilcoxon test
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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]]

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

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|Rank sum test]]; [[Mann–Whitney test|Mann–Whitney test]]). See also [[Van der Waerden test|van der Waerden test]]; [[Rank test|Rank test]].

====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>

====Comments====

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR>
</table>