Walter Schneller | Searcharxiv

arXiv2026

Edgeworth Expansions for Linear Rank Statistics -- Consolidated Version

An Edgeworth expansion of first order is established for general linear rank statistics under the null hypothesis with a remainder term that is usually of order $n^{-1}$. Furthermore, corresponding results for the second order are formulated, but not proved here. The proof for the first order is based on Stein's method and on an extension of the combinatorial method of Bolthausen. It is also shown that conditions of van Zwet imply up to a small factor our conditions for the validity of Edgeworth expansions. Moreover, our proof for the first order also provides us with a result about Edgeworth expansions for smooth functions.

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arXiv2025

Edgeworth Expansions for Linear Rank Statistics Using Stein's Method

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.

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