A uniform saddlepoint expansion for the null-distribution of the Wilcoxon- Mann-Whitney statistic

The authors give the exact coefficient of 1/N in a saddlepoint approximation to the Wilcoxon-Mann-Whitney null-distribution. This saddlepoint approximation is obtained from an Edgeworth approximation to the exponentially tilted distribution. Moreover, the rate of convergence of the relative error is uniformly of order O(1/N) in a large deviation interval as defined in Feller (1971). The proposed method for computing the coefficient of 1/N can be used to obtain the exact coefficients of 1/N , for any i. The exact formulas for the cumulant generating function and the cumulants, needed for these results, are those of van Dantzig (1947-1950).