A Bivariate Extension of the U Statistic

Let x, y, and z be three random variables with continuous cumulative distribution functions f, g, and h. In order to test the hypothesis f = g = h under certain alternatives two statistics U, V based on ranks are proposed. Recurrence relations are given for determining the probability of a given (U, V) in a sample of l x's, m y's, n z's and the different moments of the joint distribution of U and V. The means, second, and fourth moments of the joint distribution are given explicitly and the limit distribution is shown to be normal. As an illustration the joint distribution of U, V is given for the case (l, m, n) = (6, 3, 3) together with the values obtained by using the bivariate normal approximation. Tables of the joint cumulative distribution of U, V have been prepared for all cases where l + m + n 15.