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On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other

1947/03/01 by H. B. Mann, D. R. Whitney, Douglas R. Whitney · 14,109 citations
Computer Science · Engineering · Mathematics · #Bayesian Methods and Mixture Models #Combinatorics #Cumulative distribution function #Distribution (mathematics) #Diverse Scientific and Engineering Research #Infinity #Limit (mathematics) #Mathematical analysis #Mathematics #Moment (physics) #Normal distribution #Physics #Probability density function #Random variable #Statistic #Statistical hypothesis testing #Statistics #Stochastic processes and statistical mechanics #Test statistic

paper · pdf · doi:10.1214/aoms/1177730491

published in The Annals of Mathematical Statistics 18(1), 50-60 (Institute of Mathematical Statistics)

openalex publication_date 1947/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let x and y be two random variables with continuous cumulative distribution functions f and g. A statistic U depending on the relative ranks of the x's and y's is proposed for testing the hypothesis f = g. Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis f = g the probability of obtaining a given U in a sample of n x's and m y's is the solution of a certain recurrence relation involving n and m. Using this recurrence relation tables have been computed giving the probability of U for samples up to n = m = 8. At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if m, n go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives f(x) > g(x) for every x.

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