Tests for the mean of high-dimensional data

Dietmar Ferger

arxiv: 2605.16033 · v1 · pith:ZIWCQ423new · submitted 2026-05-15 · 🧮 math.ST · stat.TH

Tests for the mean of high-dimensional data

Dietmar Ferger This is my paper

Pith reviewed 2026-05-19 18:44 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords high-dimensional mean testingbootstrap approximationsquared norm statisticHilbert space embeddingcentral limit theorem in l2asymptotic level alphacovariance-free inference

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The pith

A bootstrap test based on the scaled squared norm of the sample mean yields valid level-alpha inference for high-dimensional means without sparsity or covariance structure assumptions.

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

The paper addresses testing whether the mean of high-dimensional observations equals zero when the dimension can grow freely relative to the sample size. It introduces the statistic equal to sample size times the squared Euclidean norm of the sample mean, which sidesteps covariance matrix inversion. By embedding the data vectors into the Hilbert space of square-summable sequences and applying a new central limit theorem there, the authors establish that a bootstrap approximation to this statistic produces asymptotically correct significance levels. This holds for both fixed and growing dimensions and requires no sparsity conditions or other structural restrictions on the covariance.

Core claim

The bootstrap approximation to the distribution of V_n equals n times the squared norm of the sample mean is asymptotically valid, delivering level-alpha tests for the mean vector in both fixed and increasing dimensions through a new central limit theorem in the l2 Hilbert space, without sparsity assumptions or structural conditions on the covariance matrix.

What carries the argument

The scaled squared Euclidean norm statistic V_n combined with bootstrap resampling after embedding observations into the l2 Hilbert space.

If this is right

The procedure remains valid when dimension increases without any rate restriction relative to sample size.
Covariance matrix inversion is unnecessary for the test to achieve correct asymptotic level.
No sparsity in the mean vector or covariance is needed for the bootstrap to deliver the nominal significance level.
The same embedding and central limit theorem approach covers both fixed and growing dimensions under one set of assumptions.

Where Pith is reading between the lines

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

The norm-based approach might simplify other high-dimensional problems such as testing for covariance equality.
If the l2 central limit theorem extends to weakly dependent observations, the test could apply to high-dimensional time series.
Similar bootstrap procedures on norm statistics could produce confidence regions for the mean without additional assumptions.

Load-bearing premise

The data vectors can be treated as elements of the square-summable sequence space so the new central limit theorem applies to their normalized averages.

What would settle it

A concrete high-dimensional dataset or simulation where the bootstrap procedure rejects the true null hypothesis at a rate clearly exceeding the nominal alpha level would refute the asymptotic level claim.

read the original abstract

We consider the problem of testing the mean of high-dimensional data when the dimension may grow without explicit rate restrictions relative to the sample size. The proposed procedure is based on the statistic V_n = n||Xn||^2, which avoids inversion of the covariance matrix and is therefore suitable for high-dimensional settings.We establish asymptotic distributional results for both fixed and increasing dimension by embedding the observations into the Hilbert space l2. Furthermore, we prove the asymptotic validity of a bootstrap approximation for the distribution of the test statistic. The resulting bootstrap test yields asymptotic level-a procedures without requiring sparsity assumptions or structural conditions on the covariance matrix. In all this, a new Central Limit Theorem in l2 is proving to be an extremely useful tool.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper offers a bootstrap test for high-dimensional means via an l2-norm statistic that avoids covariance inversion, but its claim of no structural conditions looks overstated given the trace-class requirement for the l2 setting.

read the letter

The main takeaway is that this paper gives a way to test the mean for high-dimensional data by using V_n = n ||X_n||^2 and a bootstrap approximation, claiming it works without sparsity or structural conditions on the covariance by embedding everything in l2. It is new in combining this particular statistic with the l2 approach and proving bootstrap validity for growing dimensions. The method avoids inverting the covariance, which is a plus in high-dim regimes where that is unstable or impossible. The paper does well in laying out asymptotic results for both fixed and increasing dimension cases. It positions the new CLT in l2 as a key tool, which could be handy if the details check out. The soft spot is the claim about no structural conditions. As the stress test notes, for a centered Gaussian in l2 to exist properly, the covariance operator must be trace-class. That imposes E[||X||^2] < infinity in a summable way, which is a condition on the covariance. The abstract says the test works without such conditions, but the framework itself seems to require it. This might be a minor oversight in wording rather than a fatal flaw, but it needs clarification. The full proofs would help assess how rigorous the new CLT is. This kind of work is for people doing theoretical statistics on high-dimensional problems, like in multivariate analysis or functional data. Readers who need practical tests for means when p is large could get some value, especially if they can verify the conditions hold for their data. It deserves a serious referee because the core idea addresses a real issue in the field, even if some technical points need tightening. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a test for the mean of high-dimensional data (dimension possibly growing with sample size) based on the statistic V_n = n ||X_n||^2. Observations are embedded into the Hilbert space l^2, a new Central Limit Theorem in l^2 is invoked to obtain limiting distributions for both fixed and increasing dimension, and asymptotic validity of a bootstrap approximation is established. The resulting test is claimed to achieve asymptotic level-α without sparsity assumptions or structural conditions on the covariance matrix.

Significance. If the asymptotic and bootstrap results hold under the stated conditions, the work would be significant for high-dimensional inference: it supplies a computationally simple, matrix-inversion-free procedure whose validity does not rely on sparsity or eigenvalue decay rates, thereby extending the range of applicable data settings. The introduction of a new CLT in l^2 as a reusable technical tool is a methodological strength that could support further infinite-dimensional extensions.

major comments (2)

[Abstract] Abstract: The central claim that the bootstrap test yields asymptotic level-α procedures 'without requiring ... structural conditions on the covariance matrix' is load-bearing for the paper's contribution. However, the l^2 embedding requires the covariance operator to be trace-class (sum of eigenvalues finite, equivalently E||X||^2 < ∞). This is a structural restriction on second moments that is not automatically satisfied by arbitrary high-dimensional data and appears to contradict the 'no structural conditions' phrasing. Please identify the section where this moment condition is stated or shown to be unnecessary for the new CLT.
[Section introducing the new CLT] The new CLT in l^2 (invoked to justify the limiting distribution of V_n and its bootstrap version) is the key technical device. Without an explicit statement of its assumptions (particularly regarding the trace-class property or moment conditions), it is impossible to verify whether the claimed absence of structural conditions is consistent with the Hilbert-space framework. A concrete counter-example or relaxation should be supplied if the CLT truly operates without trace-class covariance.

minor comments (2)

[Abstract] Abstract: 'level-a' should read 'level-α'.
[Abstract] Abstract: The sentence 'a new Central Limit Theorem in l2 is proving to be an extremely useful tool' contains a grammatical awkwardness; rephrase to 'proves to be' or 'is shown to be'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised help clarify the role of the trace-class condition in our l^2 embedding and new CLT. We address each major comment below. We agree that the basic integrability condition E||X||^2 < ∞ is required and will revise the manuscript to state all assumptions explicitly while preserving the distinction from sparsity or eigenvalue-decay restrictions.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the bootstrap test yields asymptotic level-α procedures 'without requiring ... structural conditions on the covariance matrix' is load-bearing for the paper's contribution. However, the l^2 embedding requires the covariance operator to be trace-class (sum of eigenvalues finite, equivalently E||X||^2 < ∞). This is a structural restriction on second moments that is not automatically satisfied by arbitrary high-dimensional data and appears to contradict the 'no structural conditions' phrasing. Please identify the section where this moment condition is stated or shown to be unnecessary for the new CLT.

Authors: We thank the referee for this observation. The condition E||X||^2 < ∞ (equivalently, trace-class covariance operator) is indeed required for the data to form square-integrable random elements in l^2; without it the statistic V_n would not have finite expectation and the embedding would not be valid. This is a minimal moment condition, not the kind of structural restriction (sparsity, bounded eigenvalues, or specific decay rates) we contrast against in the abstract. The condition is used from the setup in Section 2 onward and is implicit in the statement that observations are embedded into l^2. We will revise the abstract and introduction to read 'without requiring sparsity assumptions or further structural conditions on the covariance matrix beyond the basic integrability E||X||^2 < ∞' and will add an explicit reference to this moment condition in the paragraph introducing the new CLT. revision: yes
Referee: [Section introducing the new CLT] The new CLT in l^2 (invoked to justify the limiting distribution of V_n and its bootstrap version) is the key technical device. Without an explicit statement of its assumptions (particularly regarding the trace-class property or moment conditions), it is impossible to verify whether the claimed absence of structural conditions is consistent with the Hilbert-space framework. A concrete counter-example or relaxation should be supplied if the CLT truly operates without trace-class covariance.

Authors: The new CLT is formulated for random elements in the Hilbert space l^2 and therefore requires finite second moments, i.e., a trace-class covariance operator. We do not assert that the CLT holds without this condition; the Hilbert-space framework presupposes it. Consequently we cannot furnish a counter-example or relaxation demonstrating validity in its absence. In the revision we will insert a self-contained statement of the new CLT (as a theorem) that lists all assumptions explicitly, including the trace-class requirement. This will make transparent that the 'no structural conditions' claim refers only to the absence of sparsity or eigenvalue-decay assumptions beyond the necessary integrability for the l^2 setting. revision: yes

Circularity Check

0 steps flagged

Relies on new CLT in l2 presented as tool; no reduction to fitted inputs or self-referential definitions

full rationale

The paper's derivation proceeds by embedding observations into the Hilbert space l2, applying a new Central Limit Theorem in l2 to obtain the limiting distribution of V_n = n||X_n||^2 (and bootstrap version), and concluding asymptotic validity of the level-α bootstrap test. This chain does not reduce any claimed prediction to a fitted parameter by construction, nor does it invoke self-citations as load-bearing premises for uniqueness or ansatz. The central claim of procedures without sparsity or structural conditions on the covariance is presented as following from the l2 asymptotic framework rather than circular self-definition. The derivation remains self-contained against the stated external asymptotic theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on embedding data into l2 and invoking a new Central Limit Theorem in that space to obtain limiting distributions; no free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption A new Central Limit Theorem in l2 holds for the embedded high-dimensional observations.
Invoked to establish asymptotic distributional results for the test statistic under both fixed and increasing dimension.

pith-pipeline@v0.9.0 · 5634 in / 1030 out tokens · 45788 ms · 2026-05-19T18:44:09.298081+00:00 · methodology

discussion (0)

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 washburn_uniqueness_aczel unclear

?

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

We establish asymptotic distributional results ... by embedding the observations into the Hilbert space l2 ... new Central Limit Theorem in l2 ... without requiring sparsity assumptions or structural conditions on the covariance matrix.
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

?

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

finiteness of trace(Γ) as required in (3) ensures the existence of N(0,Γ) in l2

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 1 internal anchor

[1]

Araujo and E

A. Araujo and E. Gin´ e,The Central Limit Theorem for Real and Banach Valued Random Variables, New York: John Wiley & Sons, 1980

work page 1980
[2]

Bai and H

Z. Bai and H. Saranadasa,Effect of high dimension: by an example of a two sample problem, Statist. Sinica6(1996), 311–329

work page 1996
[3]

Billingsley,Convergence of Probability Measures, New York: John Wiley & Sons, 1968

P. Billingsley,Convergence of Probability Measures, New York: John Wiley & Sons, 1968

work page 1968
[4]

T. T. Cai, W. Liu and Y. Xia,Two-sample test of high dimensional means under dependence, J. R. Statist. Soc. B76Part 2 (2014), 349–372

work page 2014
[5]

Chakraborty and P

A. Chakraborty and P. Chaudhuri,Tests for high-dimensional data based on means, spatial signs and spatial ranks, Ann. Statist.45(2) (2017), 771–799

work page 2017
[6]

Chen and Y.-L

S.X. Chen and Y.-L. Qin,A two-sample test for high-dimensional data with applications to gene-set testing, Ann. Statist.38(2) (2010) 808–835. 15

work page 2010
[7]

Donoho and J

D. Donoho and J. Jin,Higher criticism for large-scale inference, especially for rare and weak effects, Statist. Sci.30(1) (2015), 1–25

work page 2015
[8]

Y. He, G. Xu, C. Wu and W. Pan,Asymptotically independent U-statistics in high-dimensional testing, Ann. Statist.49(1) (2021), 154–181

work page 2021
[9]

Henze,Asymptotische Stochastik: Eine Einf¨ uhrung mit Blick auf die Statistik, Heidelberg: Springer Nature, 2022

N. Henze,Asymptotische Stochastik: Eine Einf¨ uhrung mit Blick auf die Statistik, Heidelberg: Springer Nature, 2022

work page 2022
[10]

Huang,Projection Test for High-Dimensional Mean Vectors with Optimal Direction, Ph.D

Y. Huang,Projection Test for High-Dimensional Mean Vectors with Optimal Direction, Ph.D. dissertation, The Pennsylvania State University at University Park, 2015

work page 2015
[11]

A More Powerful Two-Sample Test in High Dimensions using Random Projection

M. Lopes, L. Jacob and M. J. Wainwright,A more powerful two-sample test in high dimensions using random projection, in: Advances in Neural Informa- tion Processing Systems (2011), pp. 1206–1214, Longer version: arXiv preprint arXiv:1108.2401

work page internal anchor Pith review Pith/arXiv arXiv 2011
[12]

G. R. Shorack,Probability for Statisticians, New York: Springer-Verlag, 2000

work page 2000
[13]

Srivastava and M

M.S. Srivastava and M. Du,A test for the mean vector with fewer observations than the dimension, J. Multivariate Anal.99(3) (2008), 386–402

work page 2008
[14]

G¨ anssler and W

P. G¨ anssler and W. Stute,Wahrscheinlichkeitstheorie, Berlin, Heidelberg, Ger- many: Springer-Verlag, 1977

work page 1977
[15]

Thulin,A high-dimensional two-sample test for the mean using random subspaces, Comput

M. Thulin,A high-dimensional two-sample test for the mean using random subspaces, Comput. Statist. Data Anal.74(2014), 26–38

work page 2014
[16]

M. J. Wichura,A note on the convergence of stochastic processes, Ann. Math. Statist.42(5), 1769–1772

work page
[17]

G. Xu, L. Lin, P. Wei and W. Pan,An adaptive two-sample test for high- dimensional means, Biometrika103(3) (2016), 609–624

work page 2016
[18]

Xue and F

K. Xue and F. Yao,Distribution and correlation-free two-sample test of high- dimensional means, Ann. Statist.48(3) (2020), 1304–1328

work page 2020
[19]

Zhang, J

J.-T. Zhang, J. Guo, B. Zhou and M.-Y. Cheng,A simple two-sample test in high dimensions based onL 2-norm, J. Amer. Statist. Assoc.115(530) (2020), 1011–1027. 16

work page 2020

[1] [1]

Araujo and E

A. Araujo and E. Gin´ e,The Central Limit Theorem for Real and Banach Valued Random Variables, New York: John Wiley & Sons, 1980

work page 1980

[2] [2]

Bai and H

Z. Bai and H. Saranadasa,Effect of high dimension: by an example of a two sample problem, Statist. Sinica6(1996), 311–329

work page 1996

[3] [3]

Billingsley,Convergence of Probability Measures, New York: John Wiley & Sons, 1968

P. Billingsley,Convergence of Probability Measures, New York: John Wiley & Sons, 1968

work page 1968

[4] [4]

T. T. Cai, W. Liu and Y. Xia,Two-sample test of high dimensional means under dependence, J. R. Statist. Soc. B76Part 2 (2014), 349–372

work page 2014

[5] [5]

Chakraborty and P

A. Chakraborty and P. Chaudhuri,Tests for high-dimensional data based on means, spatial signs and spatial ranks, Ann. Statist.45(2) (2017), 771–799

work page 2017

[6] [6]

Chen and Y.-L

S.X. Chen and Y.-L. Qin,A two-sample test for high-dimensional data with applications to gene-set testing, Ann. Statist.38(2) (2010) 808–835. 15

work page 2010

[7] [7]

Donoho and J

D. Donoho and J. Jin,Higher criticism for large-scale inference, especially for rare and weak effects, Statist. Sci.30(1) (2015), 1–25

work page 2015

[8] [8]

Y. He, G. Xu, C. Wu and W. Pan,Asymptotically independent U-statistics in high-dimensional testing, Ann. Statist.49(1) (2021), 154–181

work page 2021

[9] [9]

Henze,Asymptotische Stochastik: Eine Einf¨ uhrung mit Blick auf die Statistik, Heidelberg: Springer Nature, 2022

N. Henze,Asymptotische Stochastik: Eine Einf¨ uhrung mit Blick auf die Statistik, Heidelberg: Springer Nature, 2022

work page 2022

[10] [10]

Huang,Projection Test for High-Dimensional Mean Vectors with Optimal Direction, Ph.D

Y. Huang,Projection Test for High-Dimensional Mean Vectors with Optimal Direction, Ph.D. dissertation, The Pennsylvania State University at University Park, 2015

work page 2015

[11] [11]

A More Powerful Two-Sample Test in High Dimensions using Random Projection

M. Lopes, L. Jacob and M. J. Wainwright,A more powerful two-sample test in high dimensions using random projection, in: Advances in Neural Informa- tion Processing Systems (2011), pp. 1206–1214, Longer version: arXiv preprint arXiv:1108.2401

work page internal anchor Pith review Pith/arXiv arXiv 2011

[12] [12]

G. R. Shorack,Probability for Statisticians, New York: Springer-Verlag, 2000

work page 2000

[13] [13]

Srivastava and M

M.S. Srivastava and M. Du,A test for the mean vector with fewer observations than the dimension, J. Multivariate Anal.99(3) (2008), 386–402

work page 2008

[14] [14]

G¨ anssler and W

P. G¨ anssler and W. Stute,Wahrscheinlichkeitstheorie, Berlin, Heidelberg, Ger- many: Springer-Verlag, 1977

work page 1977

[15] [15]

Thulin,A high-dimensional two-sample test for the mean using random subspaces, Comput

M. Thulin,A high-dimensional two-sample test for the mean using random subspaces, Comput. Statist. Data Anal.74(2014), 26–38

work page 2014

[16] [16]

M. J. Wichura,A note on the convergence of stochastic processes, Ann. Math. Statist.42(5), 1769–1772

work page

[17] [17]

G. Xu, L. Lin, P. Wei and W. Pan,An adaptive two-sample test for high- dimensional means, Biometrika103(3) (2016), 609–624

work page 2016

[18] [18]

Xue and F

K. Xue and F. Yao,Distribution and correlation-free two-sample test of high- dimensional means, Ann. Statist.48(3) (2020), 1304–1328

work page 2020

[19] [19]

Zhang, J

J.-T. Zhang, J. Guo, B. Zhou and M.-Y. Cheng,A simple two-sample test in high dimensions based onL 2-norm, J. Amer. Statist. Assoc.115(530) (2020), 1011–1027. 16

work page 2020