Mann-Whitney test

A statistical test for testing the hypothesis $ H _ {0} $ of homogeneity of two samples $ X _ {1} \dots X _ {n} $ and $ Y _ {1} \dots Y _ {m} $, all $ m + n $ elements of which are mutually independent and have continuous distributions. This test, suggested by H.B. Mann and D.R. Whitney [1], is based on the statistic

$$ U = W - \frac{1}{2} m ( m + 1 ) = \ \sum _ { i=1 } ^ { n } \ \sum _ { j=1 } ^ { m } \delta _ {ij} , $$

where $ W $ is the statistic of the Wilcoxon test intended for testing the same hypothesis, equal to the sum of the ranks of the elements of the second sample among the pooled order statistics (cf. Order statistic), and

$$ \delta _ {ij} = \ \left \{ \begin{array}{ll} 1 & \textrm{ if } X _ {i} < Y _ {j} , \\ 0 & \textrm{ otherwise } . \\ \end{array} \right .$$

Thus, $ U $ counts the number of cases when the elements of the second sample exceed elements of the first sample. It follows from the definition of $ U $ that if $ H _ {0} $ is true, then

$$ \tag{* } {\mathsf E} U = \frac{nm}{2} ,\ \ {\mathsf D} U = \frac{n m ( n + m + 1 ) }{12} , $$

and, in addition, this statistic has all the properties of the Wilcoxon statistic $ W $, including asymptotic normality with parameters (*).

References

[1]	H.B. Mann, D.R. Whitney, "On a test whether one of two random variables is statistically larger than the other" Ann. Math. Stat. , 18 (1947) pp. 50–60

Comments

Instead of Mann–Whitney test, the phrase $U$-test is also used.