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arxiv: 2512.10537 · v2 · submitted 2025-12-11 · 📊 stat.ME · stat.CO

A Bayes-Motivated Quadratic-Form Test for High-Dimensional Mean Testing

Daojiang He , Suren Xu , Jing Zhou This is my paper

Pith reviewed 2026-05-16 23:30 UTC · model grok-4.3

classification 📊 stat.ME stat.CO
keywords high-dimensional mean testingBayes factorquadratic form statistictwo-sample testasymptotic normalityheterogeneous variancesrobustness to misspecification
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The pith

A Bayes factor quadratic-form test detects mean differences in high dimensions when dimension grows linearly with sample size.

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

The paper introduces a two-sample test for equality of high-dimensional mean vectors that starts from the Bayes factor computed with non-informative priors. The construction is intended for the regime in which the number of dimensions p and the sample size n satisfy p/n approaching a positive constant. Asymptotic normality of the resulting statistic is derived, together with the limiting power under local alternatives. Simulations indicate that the test competes with existing procedures when marginal variances differ across features and when sample sizes are modest, while preserving type I error control for both sparse and dense signals and remaining stable under distributional misspecification.

Core claim

We propose a two-sample mean test based on the Bayes factor with non-informative priors, specifically designed for scenarios where the dimension p grows with the sample size n with a linear rate p/n to a constant in (0, infinity). We establish the asymptotic normality of the test statistic and the asymptotic power. Through extensive simulations, we demonstrate that the proposed test performs competitively against several existing methods, particularly when the marginal variances of the individual features are heterogeneous and when the sample size is small. Furthermore, our test remains robust under distribution misspecification and maintains a well-controlled type I error rate even in small

What carries the argument

The quadratic-form test statistic obtained as the Bayes factor under flat priors on the mean difference vector.

Load-bearing premise

The ratio of dimension to sample size converges to a fixed positive finite constant.

What would settle it

A simulation with increasing n and p held at fixed ratio in which the properly standardized test statistic fails to approach a standard normal distribution under the null would falsify the asymptotic normality claim.

read the original abstract

We propose a two-sample mean test based on the Bayes factor with non-informative priors, specifically designed for scenarios where the dimension $p$ grows with the sample size $n$ with a linear rate $p/n \to c_1 \in (0, \infty)$. We establish the asymptotic normality of the test statistic and the asymptotic power. Through extensive simulations, we demonstrate that the proposed test performs competitively against several existing methods, particularly when the marginal variances of the individual features are heterogeneous and when the sample size is small. Furthermore, our test remains robust under distribution misspecification. The proposed method not only effectively detects both sparse and non-sparse differences in mean vectors but also maintains a well-controlled type I error rate, even in small-sample scenarios. We also demonstrate the performance of our proposed test using the small round blue cell tumors (SRBCT) dataset.

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.

Referee Report

2 major / 2 minor

Summary. The paper proposes a two-sample mean test derived from the Bayes factor with non-informative priors, specifically for the high-dimensional regime where p/n → c1 ∈ (0, ∞). It establishes asymptotic normality of the resulting quadratic-form statistic under the null and derives the asymptotic power. Simulations indicate competitive performance against existing methods (especially under heterogeneous marginal variances and small n), robustness to distributional misspecification, type-I error control, and detection of both sparse and dense mean differences; the method is also illustrated on the SRBCT dataset.

Significance. If the asymptotic derivations hold, the work supplies a theoretically justified Bayes-motivated quadratic statistic whose limiting behavior is explicit under linear dimension growth. The simulation evidence for small-n and heterogeneous-variance regimes, if reproducible, would address a practical gap where many existing high-dimensional tests degrade.

major comments (2)
  1. [Abstract] Abstract and simulation results: the claims of 'well-controlled type I error rate, even in small-sample scenarios' and superior performance for small n rest entirely on Monte Carlo experiments, yet all theoretical guarantees (asymptotic normality and power) are derived exclusively under p/n → c1 ∈ (0, ∞). No Berry–Esseen bounds, Edgeworth expansions, or finite-sample error analysis are supplied, so the small-n advantages may be artifacts of the chosen simulation designs rather than general properties of the procedure.
  2. [Theory] Asymptotic theory section: the derivation of the limiting null distribution of the Bayes-factor quadratic form must explicitly accommodate heterogeneous variances (the setting highlighted as advantageous in simulations). If the variance matrix is assumed diagonal or the normalization absorbs heterogeneity only under additional conditions, the stated asymptotic normality may not hold uniformly over the heterogeneous case emphasized in the abstract.
minor comments (2)
  1. [Simulations] Simulations: report the number of Monte Carlo replications, standard errors or confidence intervals for empirical type-I error and power, and any sensitivity checks to the choice of simulation parameters (e.g., variance heterogeneity levels).
  2. [Data analysis] Data analysis: clarify the preprocessing steps, feature selection, and any exclusion rules applied to the SRBCT dataset before testing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below, indicating revisions where appropriate. Our responses focus on clarifying the scope of the theoretical results and simulation evidence without overstating finite-sample guarantees.

read point-by-point responses

  1. Referee: [Abstract] Abstract and simulation results: the claims of 'well-controlled type I error rate, even in small-sample scenarios' and superior performance for small n rest entirely on Monte Carlo experiments, yet all theoretical guarantees (asymptotic normality and power) are derived exclusively under p/n → c1 ∈ (0, ∞). No Berry–Esseen bounds, Edgeworth expansions, or finite-sample error analysis are supplied, so the small-n advantages may be artifacts of the chosen simulation designs rather than general properties of the procedure.

    Authors: We agree that the small-sample claims rely on simulation evidence rather than finite-sample theory. The asymptotic results are derived under p/n → c1, and we do not claim uniform finite-sample guarantees. The simulations (covering n as small as 20–50 with p up to several hundred) show consistent type-I error control and competitive power, particularly under heterogeneity. To address the concern, we will revise the abstract to state that small-n performance is observed in simulations and add a brief discussion of simulation design robustness in the main text. revision: partial

  2. Referee: [Theory] Asymptotic theory section: the derivation of the limiting null distribution of the Bayes-factor quadratic form must explicitly accommodate heterogeneous variances (the setting highlighted as advantageous in simulations). If the variance matrix is assumed diagonal or the normalization absorbs heterogeneity only under additional conditions, the stated asymptotic normality may not hold uniformly over the heterogeneous case emphasized in the abstract.

    Authors: The derivation in Section 3 assumes a general positive-definite covariance matrix Σ with eigenvalues bounded away from 0 and ∞ (allowing heterogeneity across features). The quadratic-form statistic is normalized by consistent estimators of the diagonal elements of Σ, and the central limit theorem is applied to the resulting standardized sum under the linear growth regime. This covers the heterogeneous case without requiring diagonality. We will add an explicit remark in the theory section stating the eigenvalue bounds and confirming that the limiting normality holds uniformly under these conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; asymptotics and simulations remain independent

full rationale

The derivation establishes asymptotic normality and power of the Bayes-factor quadratic statistic under the standard high-dimensional regime p/n → c1 ∈ (0, ∞) using conventional central-limit techniques for quadratic forms. Non-informative priors are invoked in the standard manner without being fitted to the target statistic. Simulation results for small-n behavior, heterogeneous variances, and misspecification robustness are presented separately and do not feed back into the asymptotic claims. No self-citations, fitted-input renamings, or self-definitional steps appear in the load-bearing chain. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the linear high-dimensional regime and standard non-informative prior choices for the Bayes factor; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption p/n → c1 ∈ (0, ∞)
    Invoked to establish asymptotic normality and power of the test statistic.
  • standard math Non-informative priors for the Bayes factor
    Used to derive the quadratic-form test statistic.

pith-pipeline@v0.9.0 · 5449 in / 1202 out tokens · 63296 ms · 2026-05-16T23:30:38.405249+00:00 · methodology

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Reference graph

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