A uniform bound on the characteristic function of the exponentially tilted null-distribution of a simple linear rank statistic and its rate of convergence

In this paper, an upper bound of van Zwet on the characteristic function of simple linear rank statistics is, under the null-hypothesis, generalized to the characteristic function of the exponentially tilted distribution. Our present result solves a difficult and crucial step in obtaining a rate of convergence for saddlepoint expansions for more general rank statistics. In particular, it has been used by Froda and van Eeden in the proof of their saddlepoint expansion for the null-distribution of the Wilcoxon-Mann-Whitney statistic.