Edgeworth approximations for rank sum test statistics

Hettmansperger (1984) quotes a result showing that the distribution function of the Wilcoxon signed rank statistic is approximated by the usual Edgeworth series using the first four cumulants, to 0(n-1). In light of standard Edgeworth series results for random variables confined to a lattice, this result is counterintuitive. One expects correction terms to be necessary because of the lattice nature of the Wilcoxon statistic. This paper explains this apparent paradox, provides an alternative proof relying on basic Edgeworth series results, and provides a sharper result. Interesting features in this problem highlighting limitations of expansions for random variables on a lattice are discussed.