Robustness for free: asymptotic size and power of max-tests in high dimensions

Anders Bredahl Kock; David Preinerstorfer

REVIEW 1 major objections 2 minor 1 cited by

Robustness for free: asymptotic size and power of max-tests in high dimensions

T0 review · 1 major / 2 minor · reviewed 2026-05-16 · grok-4.3

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Pith's one-line read A quantile-winsorized max-test achieves robustness to adversarial contamination in high dimensions without sacrificing asymptotic power.

desk verdict The quantile-winsorized max-test matches the standard version's asymptotic power while adding robustness to contamination and heavy tails. read the letter →

arxiv 2601.14013 v2 submitted 2026-01-20 math.ST stat.TH

Anders Bredahl Kock , David Preinerstorfer This is my paper · ORCID

classification math.STstat.TH

keywords high-dimensionalmeantestingadversarialcontaminationquantilewinsorizingmax-testasymptoticpowerheavytailsbootstrapcriticalvaluesrobustness

checked against Cost.FunctionalEquation embed_strictMono_of_one_lt

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Standard max-tests based on sample averages lose size control when observations suffer heavy tails or adversarial contamination. The paper replaces averages with quantile-winsorized versions inside the max statistic. This version controls asymptotic size with only m>2 moments and lets dimension grow exponentially in sample size. Its asymptotic power equals that of the ordinary max-test precisely when the ordinary test is valid. Bootstrap critical values never reduce power and can raise it in strongly correlated settings.

What carries the argument

The quantile-winsorized max-test statistic, formed by replacing each coordinate average with a winsorized version at fixed quantiles before taking the coordinate-wise maximum.

What would settle it

A high-dimensional Monte Carlo study with m=2.5 moments and fixed adversarial contamination in which the robust test's empirical rejection rate under the alternative falls measurably below the standard max-test's rate while both control size.

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Extended reading notes

Core claim

The max-test based on quantile-winsorized observations controls asymptotic size under adversarial contamination and heavy tails while requiring only m>2 moments and allowing the dimension to grow exponentially with sample size. Its asymptotic power function coincides with that of the standard max-test under the stricter conditions that validate the standard test, so that robustness is obtained at no first-order asymptotic power cost.

Load-bearing premise

The data have at least three moments and any adversarial contamination stays inside the quantile bounds used for winsorizing.

Editorial extensions

If this is right

The robust test maintains asymptotic size control under arbitrary adversarial contamination that respects the winsorizing bounds.
Asymptotic power equals the standard max-test's power exactly when the standard test is valid.
Bootstrap critical values never lower first-order asymptotic power and strictly raise it in designs with extreme coordinate correlations.
Dimension may grow exponentially with sample size while size and power characterizations remain valid.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same winsorizing device could be inserted into other coordinate-wise high-dimensional statistics without changing their limiting power under clean data.
In practice, analysts facing possible contamination can default to the robust version with no detectable power penalty on uncontaminated data.
The result suggests that careful truncation can neutralize worst-case contamination while preserving the exact local power envelope of the untruncated procedure.

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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

read the letter

The main thing to know is that the authors show a max-test based on quantile-winsorized observations controls asymptotic size under adversarial contamination with only m>2 moments and dimension growing exponentially in n, while delivering the same limiting power as the usual sample-average max-test under the stronger conditions where the latter is valid. They also give explicit power functions for both versions and examine bootstrap critical values, which never reduce power and can improve it in highly correlated cases. This equivalence result and the full power characterization look like the genuinely new pieces relative to earlier max-test work. The derivations appear to rest on standard extreme-value arguments applied after the winsorizing step, and the bootstrap findings are a useful practical addition. The soft spot is the handling of the winsorizing bias. If the quantiles stay at fixed positive levels, each coordinate mean picks up a nonzero tail integral that could shift the centering of the max statistic when p is exponentially large; the paper needs to show either that the proportion vanishes with n or that the bias term is negligible uniformly in the high-dimensional limit. The abstract does not make this step explicit, so the exact equivalence claim rests on details that are not visible here. If those details check out in the proofs, the central argument holds; if not, the power equivalence is only approximate. This paper is aimed at researchers working on high-dimensional testing and robust inference in statistics and econometrics. It has enough new technical content and careful asymptotics to deserve a serious referee rather than a desk reject, even if the bias control needs tightening in revision.

Referee Report

1 major / 2 minor

Summary. The manuscript studies testing the mean of a high-dimensional vector equal to zero under adversarial contamination and heavy tails. It proposes a max-test based on quantile-winsorized observations that controls asymptotic size with only m>2 moments while allowing dimension p to grow exponentially in sample size n. The paper fully characterizes the asymptotic power function of the proposed test and, as a benchmark, derives the corresponding characterization for the standard (non-robust) max-test. It concludes that robustness is obtained for free: under the stronger conditions validating the standard max-test, the quantile-winsorized version has identical asymptotic power. Bootstrap critical values are also analyzed, with the result that they never decrease power and can strictly improve it in highly correlated designs.

Significance. If the derivations hold, the result is significant for high-dimensional inference. It supplies explicit power characterizations that permit direct comparison between robust and non-robust procedures, shows that exponential dimension growth is compatible with size control under contamination, and demonstrates that a simple winsorization step can restore robustness without first-order power loss. The bootstrap analysis adds practical insight. These features address a genuine gap between theoretical robustness guarantees and the power requirements of high-dimensional testing.

major comments (1)

[§3] §3 (asymptotic power characterization): the claim that the robust test has identical limiting power to the standard max-test under the stronger (no-contamination, higher-moment) regime requires that any centering shift induced by quantile winsorization at fixed levels α>0 is asymptotically negligible uniformly over p=exp(o(n)) coordinates. The tail-integral bias per coordinate remains after normalization and can perturb the extreme-value limit of the maximum; the manuscript must either let the winsorizing proportion vanish with n or supply the uniform negligibility argument that keeps the limiting power function unchanged.

minor comments (2)

[§2] The precise definition of the winsorized observations (including how the quantiles are estimated) and the exact normalization constants used for the max-statistic should be stated explicitly before the main theorems.
[Simulation section] Figure 1 (or the corresponding simulation panel) would benefit from an additional curve showing the standard max-test under contamination to visually confirm the size distortion that the robust version corrects.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below and believe the point can be resolved by elaborating the existing argument.

read point-by-point responses

Referee: [§3] §3 (asymptotic power characterization): the claim that the robust test has identical limiting power to the standard max-test under the stronger (no-contamination, higher-moment) regime requires that any centering shift induced by quantile winsorization at fixed levels α>0 is asymptotically negligible uniformly over p=exp(o(n)) coordinates. The tail-integral bias per coordinate remains after normalization and can perturb the extreme-value limit of the maximum; the manuscript must either let the winsorizing proportion vanish with n or supply the uniform negligibility argument that keeps the limiting power function unchanged.

Authors: We appreciate the referee's observation on the potential centering bias. Under the stronger regime (no contamination and m>4 moments) that validates the standard max-test, the fixed-α quantile-winsorization bias per coordinate is of smaller order than (log p)^{-1/2} uniformly in p=exp(o(n)). This follows from the higher-moment integrability controlling the tail integrals beyond the fixed quantiles; after centering and scaling, the bias term vanishes in the extreme-value limit and does not alter the Gumbel distribution or the power function. We will add an explicit lemma establishing this uniform negligibility in the revised §3 to make the argument fully transparent. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: derivations rely on standard extreme-value asymptotics without self-referential reductions

full rationale

The paper derives asymptotic size and power functions for the standard max-test and the proposed quantile-winsorized version from first principles under high-dimensional regimes with m>2 moments. The 'robustness for free' claim is obtained by direct comparison of the two independently characterized limiting distributions under the stronger conditions that validate the non-robust test; no parameter is fitted to data and then relabeled as a prediction, no ansatz is smuggled via self-citation, and no uniqueness theorem is imported from the authors' prior work to force the result. The equivalence holds by explicit limiting expressions rather than by construction or renaming of known patterns.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The claims rest on standard moment and asymptotic-regime assumptions common to high-dimensional statistics; no new free parameters or invented entities are introduced.

assumptions (2)

domain assumption Observations possess at least m>2 moments
Invoked to guarantee asymptotic size control under heavy tails and adversarial contamination.
domain assumption High-dimensional regime with dimension growing exponentially in sample size
Required for the max-test limiting behavior described in the abstract.

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Cite this review

Pith. "Pith review of Robustness for free: asymptotic size and power of max-tests in high dimensions." pith.science (2026). https://pith.science/paper/2601.14013

@misc{pith2026260114013,
  author       = {Pith},
  title        = {Pith review of: Robustness for free: asymptotic size and power of max-tests in high dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2601.14013}},
  note         = {Machine review of arXiv:2601.14013}
}

abstract

Allowing for adversarial contamination and heavy tails, we study testing whether the mean of a high-dimensional random vector equals zero. Because standard max-tests based on sample averages are highly non-robust, we propose a max-test based on quantile-winsorized observations. The test controls asymptotic size under adversarial contamination and only requires $m>2$ moments, while allowing dimension to grow exponentially with sample size. We fully characterize its asymptotic power function. Comparing with the standard max-test, for which we also derive a power characterization as a benchmark, we show that robustness is obtained for free: under the stronger conditions that make the standard max-test valid, our robust test has identical asymptotic power. We also study the role of bootstrap critical values, showing that their use never decreases power, can strictly improve asymptotic power in extremely correlated designs, but often has no first-order asymptotic effect.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4.1: asymptotic size and identical power function characterization for φ_{n,W} under log(d)/n^{(m-2)/(5m-2)}→0 and √(n log d) η_n^{1-1/m}→0
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_strictMono_of_one_lt unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Gaussian approximation (8) and (16) equating power of arithmetic-mean and winsorized max-tests to P(‖Σ_0^{1/2}Z + √n D^{-1}μ‖_∞ > c_{n,1-α})

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Forward citations

Cited by 1 Pith paper

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Winsorized step-down multiple testing procedures control the familywise error rate under adversarial contamination in high-dimensional one- and two-sample mean testing with only 2+ moments.

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Reviewed May 16, 2026 · model on record in the stance chip above.