# Wilcoxon test

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=51556