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

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.