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Edgeworth Expansions for Linear Rank Statistics Using Stein's Method

Walter Schneller

TL;DR

This work develops a systematic Stein-analytic framework to derive first- and second-order Edgeworth expansions for linear rank statistics under the null hypothesis. By combining Stein's method with Bolthausen-style combinatorics and carefully smoothing distributional comparisons, it yields explicit Edgeworth expansions in terms of standardized matrix statistics, with sharp, easily verifiable conditions reminiscent of the iid-sum literature. The paper provides comprehensive moment formulas, matrix-truncation techniques, and permutation-constructive couplings to manage dependencies, delivering practical bounds for both exact and approximating score settings. Its results extend classical Edgeworth theory to a broad class of rank-statistic problems, offering precise rates of convergence and applicability to nonparametric tests. The methods have potential impact on accuracy of critical values and p-values in rank-based inference when permutation-based null distributions are used.

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.

Edgeworth Expansions for Linear Rank Statistics Using Stein's Method

TL;DR

This work develops a systematic Stein-analytic framework to derive first- and second-order Edgeworth expansions for linear rank statistics under the null hypothesis. By combining Stein's method with Bolthausen-style combinatorics and carefully smoothing distributional comparisons, it yields explicit Edgeworth expansions in terms of standardized matrix statistics, with sharp, easily verifiable conditions reminiscent of the iid-sum literature. The paper provides comprehensive moment formulas, matrix-truncation techniques, and permutation-constructive couplings to manage dependencies, delivering practical bounds for both exact and approximating score settings. Its results extend classical Edgeworth theory to a broad class of rank-statistic problems, offering precise rates of convergence and applicability to nonparametric tests. The methods have potential impact on accuracy of critical values and p-values in rank-based inference when permutation-based null distributions are used.

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.

Paper Structure

This paper contains 36 sections, 5 equations.

Table of Contents

  1. Notations and preliminary remarks
  2. Edgeworth expansions of first order for sums of independent and identically distributed random variables
  3. Introduction and results
  4. Transition to smooth functions
  5. The central step
  6. Deducing the correct expansion
  7. The estimate of $R(q_{z})$
  8. Some analytical tools
  9. Elementary properties of the solution $f_{k}$ of Stein's equation for the function ${|x|^{k}\,q(x)}$
  10. A relation between the Hermite polynomials and the functions $q(x)$ and $f(x)$
  11. Introduction of several smooth functions
  12. Some results on multiple differences and interpolating polynomials
  13. Edgeworth expansions for linear rank statistics using Stein's method
  14. Descriptions and results
  15. Higher moments of linear rank statistics
  16. ...and 21 more sections