The normal, Edgeworth, saddlepoint and uniform approximations to the Wilcoxon–Mann–Whitney null-distribution: a numerical comparison

In the present paper, we consider four approximations to the null-distribution of the two-sample Wilcoxon–Mann–Whitney statistic, namely a normal, an Edgeworth and a saddlepoint approximation, as well as an approximation by the sum of independent uniform random variables. We make numerical comparisons of these approximations for moderate sample sizes, namely for m = 20 and 20 ≤ n ≤ 80. It turns out that the saddlepoint improves on the Edgeworth and the uniform approximations only very far in the tails, while the Edgeworth outperforms the other three for less extreme cases. We also discuss the practical importance of our results in the era of statistical packages.