Edgeworth Expansions for Linear Rank Statistics Using Stein's Method

Abstract

Edgeworth expansions of first and second order are established for general linear rank statistics under the null hypothesis with asymptotically ''sufficiently'' small remainder terms. The methods used are the Stein method combined with an extension of a combinatorial method of Bolthausen (1984). The conditions obtained for the validity of these Edgeworth expansions are very similar to the necessary and sufficient conditions found by Bickel and Robinson (1982) for the case of sums of iid random variables. But these conditions are often difficult to prove directly. For simple linear rank statistics, however, it is possible to use a result from van Zwet (1982) to verify these assumptions. Thus, we obtain conditions for the validity of Edgeworth expansions, which on the one hand are very easy to prove and on the other hand are much more general than all previously known conditions. Finally, this result is applied to the special case of approximating and exact scores.