The Wilcoxon–Mann–Whitney (WMW) test was proposed by Frank Wilcoxon in 1945 (“Wilcoxon rank sum test”) and by Henry Mann and Donald Whitney in 1947 (“Mann–Whitney U test”). However, the test is older: Gustav Deuchler introduced it in 1914 (see Kruskal 1957). Nowadays, this test is a commonly used nonparametric test for the two-sample location problem. As with many other nonparametric tests, this is based on ranks rather than on the original observations.
The sample sizes of the two groups or random samples are denoted by n and m. The observations within each sample are independent and identically distributed, and we assume independence between the two samples. The null hypothesis, H 0, is one of no difference between the two groups.
Let F and G be the distribution functions corresponding to the two samples. Then we have the null hypothesis H 0 : F(t) = G(t) for every t. Under the two-sided alternative there is a difference between F and G. Often, it is assumed that F and Gare...
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References and Further Reading
Brunner E, Munzel U (2000) The nonparametric Behrens-Fisher problem: asymptotic theory and a small sample approximation. Biom J 42:17–25
Brunner E, Munzel U (2002) Nichtparametrische Datenanalyse. Springer, Berlin
Hodges JL, Lehmann EL (1956). The efficiency of some nonparametric competitors of the t-test. Ann Math Stat 27:324–335
Hollander M, Wolfe DA (1999) Nonparametric statistical methods, 2nd edn. Wiley, New York
Kruskal WH (1957) Historical notes on the Wilcoxon unpaired two-sample test. J Am Stat Assoc 52:356–360
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Neuhäuser, M. (2011). Wilcoxon–Mann–Whitney Test. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_615
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