Edgeworth Expansions for Linear Rank Statistics Using Stein's Method

TL;DR

This paper develops Edgeworth expansions for linear rank statistics using Stein's method, providing more general and easily verifiable conditions for their validity compared to previous approaches.

Contribution

It introduces a novel combination of Stein's method with combinatorial techniques to establish Edgeworth expansions for linear rank statistics under broad conditions.

Findings

Derived first and second order Edgeworth expansions with small remainder terms.

Provided easily verifiable conditions for the validity of these expansions.

Applied results to approximation and exact scores in rank statistics.

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.