27. Wilcoxon two-sample test. Let Vi) = 1 or 0 as X; < lj or X; > lj, and let V = EEU,1 be the number of pairs X;. lj with X; < lj. (i) Then V = ES, - tn(n + 1), where SI < . . . < S; are the ranks of the y's, so that the test with rejection region V> C is equivalent to the Wilcoxon test. (ii) Any given arrangement of x 's and y 's can be transformed into the arrangement x . .. xy . . . y through a number of interchanges of neighboring elements. The smallest number of steps in which this can be done for the observed arrangement is mn - V.