Edgeworth Expansions for Linear Rank Statistics -- Consolidated Version

Walter Schneller

REVIEW 3 major objections 6 minor 22 references

First-order Edgeworth expansions for linear rank statistics proved via Stein's method

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load-bearing objection Consolidated republication of the 1989 AOS paper with a corrected distribution definition; first-order Edgeworth expansion proofs are sound but depend on a condition only verified for product-form matrices. the 3 major comments →

arxiv 2607.05284 v1 pith:H43QDT6U submitted 2026-07-06 math.ST stat.TH

Edgeworth Expansions for Linear Rank Statistics -- Consolidated Version

Walter Schneller This is my paper

classification math.ST stat.TH MSC 60F0562E20

keywords Edgeworth expansionlinear rank statisticStein's methodcombinatorial central limit theoremBolthausen combinatorial methodvan Zwet conditionspermutation distributionsecond difference condition

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the distribution of a linear rank statistic T_A = Σ_i â_{iπ(i)} (where π is a uniform random permutation and A is an n×n matrix) admits a first-order Edgeworth expansion: the distribution function F_A can be approximated by a normal density plus a cubic correction term e_{1,A}, with error bounded by a constant times D_A² = δ_A/n, which is typically of order n^{-1}. The proof combines Stein's method for normal approximation with an extension of Bolthausen's combinatorial method. The key innovation is a construction of four coupled random permutations π_1, π_2, π_3, π_4 derived from a single uniform permutation, together with a random index vector I = (I_1,...,I_8), such that successive differences ΔT_k = T_{k+1} - T_k are partially independent of earlier terms. This partial independence, achieved through a sequence of block exchanges, substitutes for the full independence available in the i.i.d. setting and allows Taylor expansions of the Stein equation to be carried out. The main theorem (Theorem 2.4) requires a bounded second-difference condition (2.5) on the distribution functions of submatrices of A; the author verifies this condition for the important special case a_{ij} = e_i d_j under van Zwet's conditions, yielding an expansion with a (log n)² factor. A companion result (Theorem 2.1) gives an Edgeworth expansion for expectations E(q(T_A)) of smooth test functions q without any smoothness condition on the distribution of T_A itself.

Core claim

The central mechanism is a combinatorial coupling that creates a chain of four permutations π_1 → π_2 → π_3 → π_4, each obtained from the previous by composing with a fixed permutation determined by a random index vector I. The successive increments ΔT_1, ΔT_2, ΔT_3 decompose T_A into pieces with controlled dependence: (I_1, J_1) is independent of π_3, (I_1, I_2, J_1, J_2) is independent of π_2, and (I_1,...,I_4, J_1,...,J_4) is independent of π_1. This layered independence suffices to turn Stein's identity E(q(T_A)) - Φ(q) = E(f'(T_A)) - E(T_A f(T_A)) into a Taylor-expansion argument where each remainder term involves only lower-order dependence, yielding the basic equation (3.20) from Thee

What carries the argument

The basic equation (3.20): E(q(T_A)) - Φ(q) + (1/2)E(T_A³)·E(T_A f'(T_A)) = R(q), where R(q) is a remainder expressible in terms of second derivatives of the Stein solution f and the increments ΔT_k. The cubic moment E(T_A³)/n captures the first-order Edgeworth correction λ_{1,A}, and bounding R(q) by D_A² gives the O(n^{-1}) rate.

Load-bearing premise

The main theorem for distribution functions (Theorem 2.4) requires condition (2.5): that the second differences of the distribution functions of all 8-row-and-column-deleted submatrices of the centered matrix Â are bounded by C₁(D_A² + y²). This condition is natural and nearly necessary, but the paper only verifies it for the special case a_{ij} = e_i d_j (via van Zwet's conditions); for general matrices A it remains an abstract assumption whose practical verifiability is unc

What would settle it

If a matrix A with σ_A > 0 could be constructed where condition (2.5) fails — i.e., where second differences of F_B grow faster than D_A² + y² for some submatrix B ∈ N(8, Â) — and the Edgeworth expansion error ||F_A - e_{1,A}|| were simultaneously observed to exceed K·D_A² for all constants K, this would show the condition is not merely sufficient but necessary.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The method provides a template for Edgeworth expansions in other combinatorial probability settings where full independence is absent but partial independence can be manufactured through coupling constructions.
The second-order expansions (Theorem 2.7), while stated without proof here, suggest that the same combinatorial framework extends to higher-order corrections if one extends the permutation chain to length 5 and uses 16-element index vectors.
The connection to van Zwet's conditions shows that the abstract second-difference condition (2.5) is satisfiable under standard moment and anti-concentration assumptions on score functions, making the expansion applicable to common rank-test statistics.
The smooth-function result (Theorem 2.1) could be used to refine Berry-Esseen-type bounds for expectations of smooth functions of rank statistics, which arise in power calculations for rank tests.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If condition (2.5) could indeed be verified for the full matrix Â alone rather than for all submatrices B ∈ N(8, Â) — as the author conjectures in Remark 2.11(b) — the result would become directly applicable to arbitrary matrices A without case-by-case verification, substantially broadening its practical reach.
The (log n)² factor in Theorem 2.12(a) arises from the characteristic-function bound in the intermediate frequency range b₃ log n ≤ |t| ≤ b₄ n^{3/2}; if sharper characteristic-function estimates were available (e.g., through non-trivial cancellation arguments), this logarithmic penalty might be removable, matching the optimal O(n^{-1}) rate.
The combinatorial coupling construction with 8-element index vectors for first order and 16-element vectors for second order suggests a general pattern: an m-th order Edgeworth expansion would require 2^{m+2}-element index vectors and a chain of m+2 permutations, with the complexity growing exponentially in the order.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Consolidated republication of the 1989 AOS paper with a corrected distribution definition; first-order Edgeworth expansion proofs are sound but depend on a condition only verified for product-form matrices.

read the letter

This is a consolidated version of Schneller's 1989 AOS paper, incorporating the corrected definition of the distribution of I from the 2026 erratum. So the core results are not new — the value here is having a corrected, self-contained version of a known result in one place. The first-order Edgeworth expansion (Theorem 2.4) is proved in full through Sections 3–6, and the argument is carefully done. The combination of Stein's method with the extended Bolthausen combinatorial construction is the right tool for this problem, and the basic equation (3.20) is derived cleanly. The connection to van Zwet's conditions (Theorem 2.12(a), proved in Section 6) is also handled well — the reduction via characteristic function estimates and the von Bahr convolution technique in Lemma 6.3 is standard but correctly executed. The constants K_1,...,K_8 are absolute, not fitted, and the expansion coefficients are defined directly from matrix entries. No circularity concerns. The author is transparent about what is proved and what is deferred. The stress-test concern about condition (2.5) is real but correctly diagnosed by the reader. The condition requires bounded second differences of F_B for all submatrices B in N(8, Â), and this is structurally required: in the proof of Lemma 5.3, conditioning on (I,J) produces an (n-8)×(n-8) submatrix, and the bound on A_3 invokes (2.5) for that B. There is no workaround within the current proof structure. The author acknowledges this in Remark 2.11(b), conjecturing that (2.5) for Â alone should suffice but admitting a proof eludes him. For product-form matrices a_{ij} = e_i d_j, Section 6 verifies (2.5) for all relevant submatrices via van Zwet's conditions. For general A, the condition remains abstract. This is not an inconsistency — the theorem is correct conditional on (2.5) holding. The gap is between the natural condition and its verifiability for general matrices. Two minor deferrals: Proposition 5.7 is only sketched (complete proof in Schneller 2025), and the second-order results (Theorem 2.7, 2.12(b)) are stated without proof. Both are less concerning — Proposition 5.7 adapts Bolthausen's established Berry-Esseen bound, and the second-order deferral is clearly flagged. This paper is for specialists in nonparametric statistics and probabilistic limit theory who need the corrected, consolidated reference. It deserves a serious referee to verify the combinatorial construction in Section 3 (especially the independence structure of Lemma 3.11) and the smoothing argument in Lemma 5.2, since these are the load-bearing pieces. The condition (2.5) issue is a known limitation, not a flaw in the proof.

Referee Report

3 major / 6 minor

Summary. This manuscript establishes a first-order Edgeworth expansion for the distribution function of general linear rank statistics T_A under the null hypothesis, with remainder O(D_A^2) = O(δ_A/n), which is typically O(n^{-1}). The proof combines Stein's method with an extension of Bolthausen's combinatorial method, yielding a basic identity (Lemma 3.20) from which both a smooth-function expansion (Theorem 2.1) and a distribution-function expansion (Theorem 2.4) are derived. The key condition (2.5) requires bounded second differences of F_B for all submatrices B in N(8, Â). For the product-form case a_{ij} = e_i d_j, the author verifies (2.5) via van Zwet's (1982) characteristic function estimates (Theorem 2.12(a)). Second-order results (Theorems 2.7, 2.12(b)) are stated but not proved, with proofs deferred to Schneller (2025). This is a consolidated version of the 1989 Annals of Statistics paper incorporating an erratum correcting the distribution of the random vector I in Section 3.

Significance. The paper provides a self-contained derivation of the first-order Edgeworth expansion for general linear rank statistics, a problem of longstanding interest in nonparametric statistics. The constants K_1,...,K_8 are absolute and not fitted to data, and the expansion coefficients λ_{1,A}, λ_{2,A} are defined directly from matrix entries, so the result is parameter-free in the sense relevant to the axioms. The connection to van Zwet's conditions (Theorem 2.12) provides a concrete, verifiable pathway for the important two-sample and general product-form cases. The transparency about the scope limitation of condition (2.5) — including the explicit conjecture in Remark 2.11(b) that the condition on submatrices B ∈ N(8, Â) should suffice for Â alone — is commendable. The erratum correction to the distribution of I (Section 3, equations (3.7)–(3.10)) is properly incorporated.

major comments (3)

[Theorem 2.4, condition (2.5); proof of Lemma 5.3, A_3 term] The load-bearing assumption (2.5) requires the second-difference bound for all B ∈ N(8, Â), and this requirement arises structurally in the proof of Lemma 5.3: when bounding A_3, the conditioning on (I, J) = (i, j) produces a submatrix B obtained by deleting 8 rows and 8 columns, and the bound |Δ²_D F_B| ≤ 2C₁D² invokes (2.5) for this B. The author is transparent about this in Remark 2.11(b). However, the practical verifiability of (2.5) for general matrices A remains unclear: it is only verified for the product-form case a_{ij} = e_i d_j in Section 6 (via van Zwet's conditions). For a general matrix A satisfying σ_A > 0, no method is provided to check (2.5). This is not an internal inconsistency — the theorem is correct conditional on (2.5) — but it limits the practical scope of Theorem 2.4. The author should add a brief discussion (perhaps 2–3 sentences in Section 2) clarifying which (
[Proposition 5.7] Proposition 5.7 is a key ingredient in the proof of Theorem 2.4 (used in (5.4) to bound A_2 in Lemma 5.3), but its proof is given only as a 'Sketch of Proof' that delegates substantial steps to Bolthausen (1984) and to Schneller (2025), Chapter 3, Section 6 (see Remark 5.9). The estimate (5.8) in particular is asserted without full justification. Since Proposition 5.7 is load-bearing for the main theorem, the sketch should be expanded to at least outline the proof of (5.8), or the reader should be given a precise pointer to where in Schneller (2025) the complete argument appears.
[Section 4, truncation argument] The truncation to |a_{ij}| ≤ 1 (used in both Sections 4 and 5) is delegated to Schneller (2025), Chapter 3, Section 4, with only a reference to Bolthausen (1984), pages 382–383 for the basic ideas. Since the truncation underlies both main theorems, a brief summary of the truncation procedure and its effect on the constants would improve self-containedness. This is a presentation gap rather than a correctness concern.

minor comments (6)

[Section 4, first line] 'We devide' should be 'We divide'.
[Section 5, Lemma 5.3 proof] The notation 'c.C₁ + c.' in the parenthetical reference to (4.2), (4.3), (4.4) is unclear; presumably this means 'c·C₁ + c·' with specific constants. Clarify.
[Lemma 6.7 proof] In the proof of Lemma 6.7, the bound on I₃ uses the substitution v = tθ and the inequality θ ≥ 1/(4n). The chain of inequalities leading to I₃ → 0 could be stated more explicitly, as the role of the moment condition (6.2) on Û is somewhat compressed.
[Section 3, Table 2] Table 2 uses the notation 'Same as π₂' and 'Same as π₃' in a way that could be confusing on first reading; a brief clarifying sentence about what 'Same' means (that the σ-field generated is the same) would help.
[References] The reference to Schneller (2025) as arXiv:2511.12187 should be checked for consistency with the arXiv identifier format used elsewhere.
[Section 2, equation (2.16)] The second inequality in (2.16) involves the exponents ((4/k)−1)₊ and ((4/s)−1)₊. It would help the reader to briefly recall (here or in Remark 2.18) under what conditions on k and s these exponents vanish, so that the rate is purely O((log n)² n⁻¹).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The three major comments all concern presentation gaps rather than correctness issues: (1) the practical verifiability of condition (2.5) for general matrices, (2) the sketch-only proof of Proposition 5.7, and (3) the truncation argument delegated to external sources. We agree with all three and will revise accordingly.

read point-by-point responses

Referee: The load-bearing assumption (2.5) requires the second-difference bound for all B in N(8, Â), and this requirement arises structurally in the proof of Lemma 5.3... For a general matrix A satisfying σ_A > 0, no method is provided to check (2.5)... The author should add a brief discussion (perhaps 2–3 sentences in Section 2) clarifying which classes of matrices beyond the product-form case satisfy (2.5).

Authors: The referee is correct that condition (2.5) is only verified for the product-form case a_{ij} = e_i d_j in Section 6, and that no general verification method is provided for arbitrary matrices. This is a genuine limitation of the practical scope of Theorem 2.4, and we agree it should be made more transparent in the manuscript itself rather than leaving the reader to infer it. We will add a brief discussion in Section 2 (after Remark 2.11) clarifying that (2.5) is currently verified only for product-form matrices via van Zwet's conditions (Theorem 2.12(a)), and that for general matrices satisfying σ_A > 0, the condition remains a structural assumption whose verifiability is an open problem. We will also note that Remark 2.11(b) already conjectures that the condition on submatrices B ∈ N(8, Â) should suffice for Â alone, which would, if true, reduce the verification burden. This addition will not change any results or proofs. revision: yes
Referee: Proposition 5.7 is a key ingredient in the proof of Theorem 2.4 (used in (5.4) to bound A_2 in Lemma 5.3), but its proof is given only as a 'Sketch of Proof' that delegates substantial steps to Bolthausen (1984) and to Schneller (2025), Chapter 3, Section 6 (see Remark 5.9). The estimate (5.8) in particular is asserted without full justification. Since Proposition 5.7 is load-bearing for the main theorem, the sketch should be expanded to at least outline the proof of (5.8), or the reader should be given a precise pointer to where in Schneller (2025) the complete argument appears.

Authors: We agree that Proposition 5.7 is load-bearing and that the current sketch, particularly regarding estimate (5.8), is too compressed. The estimate (5.8) is in fact established within the sketch itself: the bound is decomposed into B_1 and B_2, and each is bounded explicitly — B_1 via the truncation set Γ and B_2 via the conditional expectation bound E(|T_A| | π(i)=j) ≤ c_8, which follows from the same arguments as in the proof of Lemma 4.1 (cf. Bolthausen 1984, p. 385, lines 3–14). However, we concede that the logical flow is not clearly delineated, and the connection between (5.8) and the subsequent B_1, B_2 decomposition could be made more explicit. We will expand the sketch to clearly separate the derivation of (5.8) from its application, and to indicate more precisely which steps follow Bolthausen (1984) and which are original. We will also add a precise pointer to Schneller (2025), Chapter 3, Section 6, where a complete (though slightly different) proof for the case |â_{ij}| ≤ 1 is given. revision: yes
Referee: The truncation to |a_{ij}| ≤ 1 (used in both Sections 4 and 5) is delegated to Schneller (2025), Chapter 3, Section 4, with only a reference to Bolthausen (1984), pages 382–383 for the basic ideas. Since the truncation underlies both main theorems, a brief summary of the truncation procedure and its effect on the constants would improve self-containedness.

Authors: The referee is right that the truncation is a foundational step for both main theorems and that the current treatment — a single sentence referencing Schneller (2025) and Bolthausen (1984) — is insufficient for self-containedness. We will add a brief summary paragraph at the beginning of Section 4 (before Lemma 4.1) outlining the truncation procedure: the key idea is to replace large entries â_{ij} by truncated versions a'_{ij} = d(â'_{ij}) where â'_{ij} = â_{ij} 1{|â_{ij}| ≤ 1/2}, and to control the resulting approximation error. We will note that this may require reducing the constant ε₀ in the condition β_A ≤ ε₀ n, and that the effect on the constants K_1, ..., K_5 in Theorems 2.1 and 2.4 is absorbed into the absolute nature of these constants. The full details will remain referenced to Schneller (2025), Chapter 3, Section 4, but the summary will allow the reader to understand the logical role of the truncation without consulting the external source. revision: yes

Circularity Check

0 steps flagged

No significant circularity: the derivation is self-contained with absolute constants and externally verified conditions; self-citations are for deferred proofs, not load-bearing definitions.

full rationale

The paper's central result (Theorem 2.4) is derived from first principles using Stein's method and the combinatorial method of Bolthausen (1984), an external source. The expansion coefficients λ_{1,A}, λ_{2,A} are defined directly from the matrix entries (Section 2), not fitted to data. The constants K_1,...,K_8 are absolute positive constants, not parameters tuned to outputs. Condition (2.5) is a structural assumption on second differences of F_B, not a definition of the conclusion. The verification of (2.5) for the product-form case (Theorem 2.12(a), Section 6) uses van Zwet (1982), an external result, to establish the characteristic function estimate (Lemma 6.5), which is then converted to (2.5) via Lemma 6.3 — a genuine mathematical derivation, not a renaming. Self-citations to Schneller (1987/2025) appear for: (i) the truncation argument (Section 4, referencing Chapter 3, Section 4), (ii) complete proofs of the second-order result (Section 7), and (iii) Proposition 5.7's complete proof (Remark 5.9). These are deferred proofs of lemmas whose statements are given in the paper, not self-citations that define the central result. The proof of Lemma 5.3 invokes (2.5) for submatrices B ∈ N(8, Â) because the combinatorial construction produces such submatrices when conditioning on (I, J) — this is a genuine structural requirement of the proof method, not circular reasoning. The author is transparent in Remark 2.11(b) that (2.5) for Â alone might suffice but a proof 'eludes me.' This is a scope limitation, not circularity. No step in the derivation chain reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities, particles, forces, or dimensions. The random vector I = (I_1,...,I_8) and the permutations u(i), t(i), s(i) are mathematical constructions within the proof, not postulated objects requiring independent evidence. There are no free parameters fitted to data; all constants K_1,...,K_8 are absolute constants whose existence is proved. The expansion coefficients λ_{1,A}, λ_{2,A} are defined directly from matrix entries.

axioms (5)

standard math Stein's equation (3.2): f'(x) - x f(x) = q(x) - Φ(q) characterizes the standard normal distribution, and the solution (3.1) satisfies the stated bounds (4.5) from Erickson (1974).
Invoked in Section 3 as the foundation of the method. This is a standard result in Stein's method, cited to Stein (1972) and Erickson (1974).
standard math Bolthausen's Berry-Esseen bound (1.1): sup_z |F_A(z) - Φ(z)| ≤ K β_A/n for some absolute constant K > 0.
Used in Proposition 5.7 and in the proof of (5.4) to control the distribution function on intervals. This is the result of Bolthausen (1984), external to the author.
standard math van Zwet's characteristic function estimate: under conditions (2.13)–(2.15), the characteristic function of T_A satisfies |F̂_B(t)| ≤ b_1 n^{-b_2 log n} for b_3 log n ≤ |t| ≤ b_4 n^{3/2}.
Invoked in Lemma 6.5 to establish condition (2.5) for the case a_{ij} = e_i d_j. This is Theorem 2.1 of van Zwet (1982), external to the author.
domain assumption The truncation |a_{ij}| ≤ 1 can be established without loss of generality for the purposes of the proofs.
Stated at the beginning of Section 4: 'We must assume |a_{ij}| ≤ 1 for all i,j. We do not show here how this truncation is established and refer the reader to Schneller (2025), Chapter 3, Section 4.' This is a standard technique (cf. Bolthausen 1984, pp. 382–383) but the details are deferred.
domain assumption Condition (2.5) is 'almost necessary' for Edgeworth expansions, analogous to Bickel and Robinson (1982) conditions in the i.i.d. case.
Stated in Remark 2.11(a). The paper claims this condition is natural but acknowledges it is 'not very practicable' for general matrices.

pith-pipeline@v1.1.0-glm · 24165 in / 3656 out tokens · 126037 ms · 2026-07-07T19:46:24.160728+00:00 · methodology

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read the original abstract

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.

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

D.(1986)

Barbour, A. D.(1986). Asymptotic expansions based on smooth functions in the central limit theorem.Probab. Theory Related Fields72289–303. DOI: 10.1007/BF00699108

work page doi:10.1007/bf00699108 1986
[2]

Bickel, P. J. and Robinson, J.(1982). Edgeworth expansions and smoothness.Ann. Probab.10500–503. DOI: 10.1214/aop/1176993873

work page doi:10.1214/aop/1176993873 1982
[3]

Bickel, P. J. and van Zwet, W. R.(1978). Asymptotic expansions for the power of distribution-free tests in the two-sample problem.Ann. Statist.6937–1004. DOI: 10.1214/aos/1176344305

work page doi:10.1214/aos/1176344305 1978
[4]

An estimate of the remainder in a combinatorial central limit theorem.Z

Bolthausen, E.(1984). An estimate of the remainder in a combinatorial central limit theorem.Z. Wahrsch. verw. Gebiete66379–386. DOI: 10.1007/BF00533704

work page doi:10.1007/bf00533704 1984
[5]

Does, R. J. M. M.(1982). Berry-Esseen theorems for simple linear rank statistics under the null-hypothesis.Ann. Probab.10982–991. DOI: 10.1214/aop/1176993719

work page doi:10.1214/aop/1176993719 1982
[6]

Does, R. J. M. M.(1983). An Edgeworth expansion for simple linear rank statistics under the null-hypothesis.Ann. Statist.11607–624. DOI: 10.1214/aos/1176346166. 24Walter Schneller

work page doi:10.1214/aos/1176346166 1983
[7]

The Annals of Probability , author =

Erickson, R. V.(1974).L 1 bounds for asymptotic normality ofm-dependent sums using Stein’s technique.Ann. Probab.2522–529. DOI: 10.1214/aop/1176996670

work page doi:10.1214/aop/1176996670 1974
[8]

Feller, W.(1971).An Introduction to Probability Theory and Its Application2, 2nd ed

work page 1971
[9]

H´ajek, J

Wiley, New York. H´ajek, J. and ˇSid´ak, Z. and Sen, P. K.(1999).Theory of Rank Tests, 2nd ed. Academic

work page 1999
[10]

DOI: 10.1016/B978-0-12-642350-1.X5017-6

Press, San Diego and London. DOI: 10.1016/B978-0-12-642350-1.X5017-6

work page doi:10.1016/b978-0-12-642350-1.x5017-6
[11]

Ho, S. T. and Chen, L. H. Y.(1978). AnL p bound for the remainder in a combinatorial central limit theorem.Ann. Probab.6231–249. DOI: 10.1214/aop/1176995570

work page doi:10.1214/aop/1176995570 1978
[12]

and Leibler, R

Hoeffding, W.(1951). A combinatorial central limit theorem.Ann. Math. Statist.22 558–566. DOI: 10.1214/aoms/1177729545

work page doi:10.1214/aoms/1177729545 1951
[13]

On the Hoeffding’s combinatorial central limit theorem.Ann

Motoo, M.(1957). On the Hoeffding’s combinatorial central limit theorem.Ann. Inst. Statist. Math.8145–154. DOI: 10.1007/BF02863580

work page doi:10.1007/bf02863580 1957
[14]

Schwarz,Estimating the dimension of a model, The annals of statistics, 6 (1978), pp

Robinson, J.(1978). An asymptotic expansion for samples from a finite population.Ann. Statist.61005–1011. DOI: 10.1214/aos/1176344306

work page doi:10.1214/aos/1176344306 1978
[15]

Schneller, W.(1987).Edgeworth-Entwicklungen f¨ ur lineare Rangstatistiken. Ph.D. thesis, Technische Universit¨ at Berlin

work page 1987
[16]

A short proof of Motoo’s combinatorial central limit theorem using Stein’s method.Probab

Schneller, W.(1988). A short proof of Motoo’s combinatorial central limit theorem using Stein’s method.Probab. Theory Related Fields78249–252. DOI: 10.1007/BF00322021

work page doi:10.1007/bf00322021 1988
[17]

Edgeworth expansions for linear rank statistics.Ann

Schneller, W.(1989). Edgeworth expansions for linear rank statistics.Ann. Statist.17 1103–1123. DOI: 10.1214/aos/1176347258

work page doi:10.1214/aos/1176347258 1989
[18]

thesis from 1987]

Schneller, W.(2025).Edgeworth Expansions for Linear Rank Statistics Using Stein’s Method[English translation of an updated version of the author’s Ph.D. thesis from 1987]. Available at arXiv:2511.12187

work page arXiv 2025
[19]

Erratum: Edgeworth expansions for linear rank statistics.Ann

Schneller, W.(2026). Erratum: Edgeworth expansions for linear rank statistics.Ann. Statist.541649–1653. DOI: 10.1214/25-AOS2611

work page doi:10.1214/25-aos2611 2026
[20]

Supplement A to ”Erratum: Edgeworth expansions for linear rank statistics.” DOI: 10.1214/25-AOS2611SUPPA

Schneller, W.(2026a). Supplement A to ”Erratum: Edgeworth expansions for linear rank statistics.” DOI: 10.1214/25-AOS2611SUPPA

work page doi:10.1214/25-aos2611suppa
[21]

Supplement B to ”Erratum: Edgeworth expansions for linear rank statistics.” DOI: 10.1214/25-AOS2611SUPPB

Schneller, W.(2026b). Supplement B to ”Erratum: Edgeworth expansions for linear rank statistics.” DOI: 10.1214/25-AOS2611SUPPB

work page doi:10.1214/25-aos2611suppb
[22]

A bound for the error in the normal approximation to the distribution of a sum of dependent random variables.Proc

Stein, C.(1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables.Proc. Sixth Berkeley Symp. Math. Statist. Probab.2 583–602. Univ. California Press URL: https://projecteuclid.org/ebooks/berkeley- symposium-on-mathematical-statistics-and-probability/Proceedings-of-the-Sixth- Berkeley-Symposium-on...

work page doi:10.1007/bf02591678 1972

[1] [1]

D.(1986)

Barbour, A. D.(1986). Asymptotic expansions based on smooth functions in the central limit theorem.Probab. Theory Related Fields72289–303. DOI: 10.1007/BF00699108

work page doi:10.1007/bf00699108 1986

[2] [2]

Bickel, P. J. and Robinson, J.(1982). Edgeworth expansions and smoothness.Ann. Probab.10500–503. DOI: 10.1214/aop/1176993873

work page doi:10.1214/aop/1176993873 1982

[3] [3]

Bickel, P. J. and van Zwet, W. R.(1978). Asymptotic expansions for the power of distribution-free tests in the two-sample problem.Ann. Statist.6937–1004. DOI: 10.1214/aos/1176344305

work page doi:10.1214/aos/1176344305 1978

[4] [4]

An estimate of the remainder in a combinatorial central limit theorem.Z

Bolthausen, E.(1984). An estimate of the remainder in a combinatorial central limit theorem.Z. Wahrsch. verw. Gebiete66379–386. DOI: 10.1007/BF00533704

work page doi:10.1007/bf00533704 1984

[5] [5]

Does, R. J. M. M.(1982). Berry-Esseen theorems for simple linear rank statistics under the null-hypothesis.Ann. Probab.10982–991. DOI: 10.1214/aop/1176993719

work page doi:10.1214/aop/1176993719 1982

[6] [6]

Does, R. J. M. M.(1983). An Edgeworth expansion for simple linear rank statistics under the null-hypothesis.Ann. Statist.11607–624. DOI: 10.1214/aos/1176346166. 24Walter Schneller

work page doi:10.1214/aos/1176346166 1983

[7] [7]

The Annals of Probability , author =

Erickson, R. V.(1974).L 1 bounds for asymptotic normality ofm-dependent sums using Stein’s technique.Ann. Probab.2522–529. DOI: 10.1214/aop/1176996670

work page doi:10.1214/aop/1176996670 1974

[8] [8]

Feller, W.(1971).An Introduction to Probability Theory and Its Application2, 2nd ed

work page 1971

[9] [9]

H´ajek, J

Wiley, New York. H´ajek, J. and ˇSid´ak, Z. and Sen, P. K.(1999).Theory of Rank Tests, 2nd ed. Academic

work page 1999

[10] [10]

DOI: 10.1016/B978-0-12-642350-1.X5017-6

Press, San Diego and London. DOI: 10.1016/B978-0-12-642350-1.X5017-6

work page doi:10.1016/b978-0-12-642350-1.x5017-6

[11] [11]

Ho, S. T. and Chen, L. H. Y.(1978). AnL p bound for the remainder in a combinatorial central limit theorem.Ann. Probab.6231–249. DOI: 10.1214/aop/1176995570

work page doi:10.1214/aop/1176995570 1978

[12] [12]

and Leibler, R

Hoeffding, W.(1951). A combinatorial central limit theorem.Ann. Math. Statist.22 558–566. DOI: 10.1214/aoms/1177729545

work page doi:10.1214/aoms/1177729545 1951

[13] [13]

On the Hoeffding’s combinatorial central limit theorem.Ann

Motoo, M.(1957). On the Hoeffding’s combinatorial central limit theorem.Ann. Inst. Statist. Math.8145–154. DOI: 10.1007/BF02863580

work page doi:10.1007/bf02863580 1957

[14] [14]

Schwarz,Estimating the dimension of a model, The annals of statistics, 6 (1978), pp

Robinson, J.(1978). An asymptotic expansion for samples from a finite population.Ann. Statist.61005–1011. DOI: 10.1214/aos/1176344306

work page doi:10.1214/aos/1176344306 1978

[15] [15]

Schneller, W.(1987).Edgeworth-Entwicklungen f¨ ur lineare Rangstatistiken. Ph.D. thesis, Technische Universit¨ at Berlin

work page 1987

[16] [16]

A short proof of Motoo’s combinatorial central limit theorem using Stein’s method.Probab

Schneller, W.(1988). A short proof of Motoo’s combinatorial central limit theorem using Stein’s method.Probab. Theory Related Fields78249–252. DOI: 10.1007/BF00322021

work page doi:10.1007/bf00322021 1988

[17] [17]

Edgeworth expansions for linear rank statistics.Ann

Schneller, W.(1989). Edgeworth expansions for linear rank statistics.Ann. Statist.17 1103–1123. DOI: 10.1214/aos/1176347258

work page doi:10.1214/aos/1176347258 1989

[18] [18]

thesis from 1987]

Schneller, W.(2025).Edgeworth Expansions for Linear Rank Statistics Using Stein’s Method[English translation of an updated version of the author’s Ph.D. thesis from 1987]. Available at arXiv:2511.12187

work page arXiv 2025

[19] [19]

Erratum: Edgeworth expansions for linear rank statistics.Ann

Schneller, W.(2026). Erratum: Edgeworth expansions for linear rank statistics.Ann. Statist.541649–1653. DOI: 10.1214/25-AOS2611

work page doi:10.1214/25-aos2611 2026

[20] [20]

Supplement A to ”Erratum: Edgeworth expansions for linear rank statistics.” DOI: 10.1214/25-AOS2611SUPPA

Schneller, W.(2026a). Supplement A to ”Erratum: Edgeworth expansions for linear rank statistics.” DOI: 10.1214/25-AOS2611SUPPA

work page doi:10.1214/25-aos2611suppa

[21] [21]

Supplement B to ”Erratum: Edgeworth expansions for linear rank statistics.” DOI: 10.1214/25-AOS2611SUPPB

Schneller, W.(2026b). Supplement B to ”Erratum: Edgeworth expansions for linear rank statistics.” DOI: 10.1214/25-AOS2611SUPPB

work page doi:10.1214/25-aos2611suppb

[22] [22]

A bound for the error in the normal approximation to the distribution of a sum of dependent random variables.Proc

Stein, C.(1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables.Proc. Sixth Berkeley Symp. Math. Statist. Probab.2 583–602. Univ. California Press URL: https://projecteuclid.org/ebooks/berkeley- symposium-on-mathematical-statistics-and-probability/Proceedings-of-the-Sixth- Berkeley-Symposium-on...

work page doi:10.1007/bf02591678 1972