Gaussian comparison above the median

Colin B. Fogarty

Gaussian comparison above the median

T0 review · 0 major / 6 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Smaller-variance Gaussians dominate on convex sets above the median

desk verdict Clean Gaussian comparison inequality resolving two conjectures; proof is solid and self-contained read the letter →

arxiv 2607.06874 v1 pith:TAIXSCOA submitted 2026-07-08 math.ST stat.TH

Colin B. Fogarty This is my paper · ORCID

classification math.STstat.TH

keywords covariancegaussianclosedconvexleastcomparisonlargerone-sided

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

The paper proves that if X and Y are centered Gaussian vectors with Y having the larger covariance matrix in the Loewner order (Σ_Y − Σ_X positive semidefinite), then for any closed convex set K that captures at least half the probability of Y, the smaller-variance vector X assigns at least as much probability to K as Y does. This is a one-sided analogue of Anderson's Theorem: Anderson's classical result requires K to be symmetric about the origin and then gives stochastic dominance for all thresholds; this paper drops the symmetry requirement and recovers the inequality above the median threshold. The proof works by interpolating between the two covariance matrices along a heat-equation path, showing that the Gaussian probability of each translate of K is nonincreasing along that path whenever it is at least 1/2. The key structural ingredient is the Ehrhard-Borell inequality, which gives concavity of the inverse-normal-transformed probability as a function of the translation parameter, forcing the relevant Hessian to be negative semidefinite above the 1/2 threshold. As a statistical corollary, conservative covariance estimation (using an estimator that overshoots the true asymptotic variance) yields valid one-sided and order-restricted inference at significance levels below 1/2, extending a tool previously limited to symmetric two-sided testing.

What carries the argument

The proof interpolates between Σ_X and Σ_Y via Z_t = X + t^{1/2}W with Cov(W) = Σ_Y − Σ_X, so that Z_t satisfies an anisotropic heat equation ∂G_t/∂t = (1/2) tr(Δ ∇²_v G_t). The Ehrhard-Borell inequality implies Φ^{-1}(G_t(v)) is concave in v, which forces ∇²_v G_t(v) ⪯ 0 whenever G_t(v) ≥ 1/2. Since Δ ⪰ 0, the trace tr(Δ ∇²_v G_t) ≤ 0, so G_t is nonincreasing in t along the path wherever it stays above 1/2. A continuity argument shows it stays above 1/2 for all t ∈ [0,1] if it starts above 1/2 at t=1.

What would settle it

A counterexample would be: a pair of centered Gaussians with Σ_Y − Σ_X ⪰ 0 and a closed convex set K with pr(Y ∈ K) ≥ 1/2 but pr(X ∈ K) < pr(Y ∈ K). The paper itself provides a counterexample showing that replacing the Loewner order with weaker conditions (larger marginal variances plus Sudakov-Fernique increment conditions) is insufficient — there, the tail probabilities cross above the median, violating the conclusion.

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Extended reading notes

Core claim

The central discovery is that Anderson's symmetry requirement can be replaced by a median threshold: for centered Gaussians ordered by covariance, the smaller-covariance law dominates the larger-covariance law on every closed convex set whose reference probability is at least 1/2. The mechanism is that the Ehrhard-Borell inequality forces the inverse-normal transform of the translated set probability to be concave, which makes the time-derivative of the interpolated probability nonpositive whenever that probability is at least 1/2, so the probability cannot increase as one moves from the smaller to the larger covariance.

Load-bearing premise

The entire argument rests on the Ehrhard-Borell inequality providing concavity of the inverse-normal-transformed probability of translated convex sets. If that concavity fails or cannot be extended to the boundary case pr(Y ∈ K) = 1/2, the sign of the derivative along the interpolation path is uncontrolled and the comparison breaks down.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main thing to know: this paper proves a one-sided analogue of Anderson's Theorem for Gaussian measures, and in doing so resolves conjectures from Ehm and Krüger (2018) and Cohen and Fogarty (2022). Theorem 1 says that if X and Y are centered Gaussians with Σ_Y ⪰ Σ_X in Loewner order, then for any closed convex set K with pr(Y ∈ K) ≥ 1/2, we have pr(X ∈ K) ≥ pr(Y ∈ K). This is a genuine new result, not a re-derivation, and it removes the mirror symmetry requirement that limited Anderson's original theorem to two-sided inference. The Ehm-Krüger conjecture gets proved under weaker conditions than originally stated (quasiconvexity instead of convex monotonicity). The counterexample in Appendix C is a nice addition — it shows the Loewner ordering can't be replaced by marginal variance dominance plus Sudakov-Fernique conditions, which is exactly the weaker condition Cohen and Fogarty had conjectured would suffice. The proof structure is well-organized: positive definite case with strict inequality first, then boundary, then singular cases by projection. The key ingredients are the Ehrhard-Borell inequality (to get concavity of Φ^{-1}(G_t(v)) in v, which controls the Hessian sign) and an anisotropic heat equation interpolation. Both are verified explicitly in the appendix. The continuity and differentiation-under-the-integral arguments in A.3 use uniform eigenvalue bounds and look correct. The stress-test flagged the Ehrhard-Borell application as the structural linchpin, and I agree it's sound — Σ_t^{-1/2}(K+v) is convex when K is, and the chain rule in A.4 checks out. The statistical application (Proposition 1) is well-motivated and follows from standard weak convergence arguments. It's not where the mathematical interest lives, but it justifies why the theorem matters: conservative covariance estimation is common in randomized experiments and design-based inference, and this extends that framework to one-sided and order-restricted tests at conventional significance levels. Soft spots are minor. The restriction to α < 1/2 is inherent to the result and the author is upfront about it. The boundary case handling via enlargement (A.5) is clean but standard. I didn't find any load-bearing concerns. This is a solid paper that deserves a serious referee. The core theorem is clean, the proof is self-contained, and it resolves open problems in the literature.

Referee Report

0 major / 6 minor

Summary. The manuscript proves a Gaussian comparison inequality (Theorem 1) stating that for centered Gaussian vectors X, Y with covariance matrices ordered as Σ_Y − Σ_X ⪰ 0, the smaller-covariance law assigns at least as much probability as the larger-covariance law to every closed convex set K with pr(Y ∈ K) ≥ 1/2. This provides a one-sided analogue of Anderson's Theorem (Anderson, 1955), which requires mirror symmetry of the set. The proof proceeds through several stages: the positive definite case with strict inequality (Appendix A.1), the boundary case pr(Y ∈ K) = 1/2 via set enlargement (A.5), singular Σ_X via L² convergence (A.6), and singular Σ_Y via projection (A.7). The key technical ingredients are a Gaussian interpolation satisfying the anisotropic heat equation and the Ehrhard–Borell inequality, which establishes concavity of Φ⁻¹(G_t(v)) in v and hence negative semidefiniteness of the Hessian ∇²_v G_t(v) whenever G_t(v) ≥ 1/2. A corollary extends the result to upper tail probabilities of lower semicontinuous quasiconvex functions above the median, and Proposition 1 provides a statistical application justifying conservative one-sided and order-restricted inference using conservative covariance estimators at significance levels α < 1/2. A counterexample in Appendix C disproves a conjecture of Cohen and Fogarty (2022) under weaker (Sudakov–Fernique) covariance conditions.

Significance. The result fills a genuine gap between Anderson's Theorem (which requires mirror symmetry and gives full stochastic dominance) and one-sided testing needs in statistical practice. The restriction to α ≤ 1/2 is natural for hypothesis testing and does not limit practical applicability. The proof is self-contained, building on the Ehrhard–Borell inequality and Anderson's Theorem, both external to the author. The explicit verification of the anisotropic heat equation identity (A.2), the careful continuity and differentiation-under-the-integral justifications (A.3), and the chain rule computation (A.4) are commendable. The counterexample in Appendix C is a valuable contribution that sharpens understanding of why the Loewner ordering is essential. The statistical application (Proposition 1) is well-motivated and correctly handles nuisance parameter estimation and plug-in tail probability estimation, including the discontinuous case arising from chi-bar-square laws.

minor comments (6)

In Appendix A.1, the string 'Σ_t = Σ^{1/2}_t Σ^{1/2}_t' appears to state that Σ_t equals its own symmetric square root product, which is trivially true but does not define the square root. The intended statement is presumably that Σ_t admits a symmetric invertible square root, i.e., Σ_t = Σ^{1/2}_t Σ^{1/2}_t where Σ^{1/2}_t is the unique positive definite square root. A brief clarification would avoid momentary confusion.
In Corollary 1, the notation m^-_Y = inf{t : pr{f(Y) ≤ t} ≥ 1/2} defines a lower median, while Proposition 1 later uses m^+_Y = sup{t : S_ξ(t; Σ_Y) ≥ 1/2}, an upper median. The relationship between these two definitions (they coincide when the distribution is continuous) could be stated explicitly to aid readers connecting the corollary to the proposition.
In Proposition 1, the condition lim_{h→0+} S_ξ(m^+_Y + h; Σ_Y) = 1/2 ensures no downward jump from 1/2. The subsequent remark about chi-bar-square laws is helpful, but a brief explicit statement of what 'projection onto {0}' means in this context (i.e., the degenerate case where the chi-bar-square distribution is a point mass at 0) would improve accessibility for readers less familiar with order-restricted inference.
In §3, the statement 'The upper tail probabilities cross at a = 0.08' is followed by 'The median of g(Y) is m^-_Y = 0.23 > 0.08.' The logic is that the crossing occurs below the median, so Corollary 1's guarantee applies above 0.23. This is correct but the phrasing could more directly state that the crossing being below the median is precisely what Corollary 1 requires.
The reference 'Harshaw et al. (2026)' lists the journal as 'Journal of the American Statistical Association, page to appear.' If the paper has appeared by the time of publication, the reference should be updated with volume and page numbers.
In Appendix A.3, the bounds C_V, C_{V,2}, C_{V,3}, C_{V,4} are introduced without explicit definitions. While their existence follows from the surrounding inequalities, stating them as explicit constants (or at least noting they depend only on V, m, M, and d) would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the proof is self-contained with external mathematical foundations

full rationale

The paper's central result (Theorem 1) is proved from first principles using the Ehrhard-Borell inequality (Ehrhard 1983, Borell 2003) and Anderson's Theorem (Anderson 1955), both external to the author. Every intermediate step is explicitly verified: the anisotropic heat equation identity is checked by direct computation (A.2), the chain rule for the Hessian is justified (A.4), continuity and differentiation under the integral are established with uniform bounds (A.3), and the boundary/singular cases are handled by standard approximation and projection arguments (A.5-A.7). The self-citation to Cohen and Fogarty (2022) is not load-bearing—it references a conjecture that the present paper partially proves (under stronger conditions) and partially disproves (via the explicit counterexample in Appendix C). Proposition 1 follows from Theorem 1 via standard weak convergence and continuous mapping arguments, not from any fitted or self-referential premise. No step in the derivation chain reduces to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities or free parameters. The proof relies on standard mathematical tools (Ehrhard-Borell inequality, dominated convergence, Portmanteau theorem) and external prior results (Anderson's Theorem). All covariance matrices in examples are explicitly specified.

assumptions (4)

standard math Ehrhard-Borell inequality: Φ^{-1}(γ_d(λA + (1-λ)B)) ≥ λΦ^{-1}(γ_d(A)) + (1-λ)Φ^{-1}(γ_d(B)) for convex sets A, B
Invoked in Appendix A.1 to establish concavity of Φ^{-1}(G_t(v)) in v, which is the key step in showing the Hessian ∇²_v G_t(v) ⪯ 0.
standard math Anderson's Theorem (Anderson 1955)
Cited as the symmetric precursor result; the present paper provides the one-sided analogue.
standard math Dominated convergence theorem for exchanging integration and differentiation
Used in Appendix A.3 to justify differentiation under the integral sign for G_t(v) and its partial derivatives.
standard math Portmanteau theorem for weak convergence
Used in Appendix A.6 to handle the singular Σ_X case via L² convergence of Z_t to X.

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Pith. "Pith review of Gaussian comparison above the median." pith.science (2026). https://pith.science/paper/TAIXSCOA

@misc{pith2026260706874,
  author       = {Pith},
  title        = {Pith review of: Gaussian comparison above the median},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TAIXSCOA}},
  note         = {Machine review of arXiv:2607.06874}
}

read the original abstract

We prove a Gaussian comparison inequality for closed convex sets with reference probability at least 1/2. For centered Gaussian vectors whose covariance matrices are ordered in the Loewner sense, the smaller covariance law assigns at least as much probability as the larger covariance law to every closed convex set with measure at least 1/2 under the larger covariance. This provides a one-sided analogue of Anderson's Theorem for Gaussian measures. As a statistical application, the result justifies one-sided and order-restricted inference using conservative covariance estimators at significance levels below 1/2 for test statistics whose acceptance regions are closed and convex but not necessarily symmetric.

Figures

Figures reproduced from arXiv: 2607.06874 by the authors.

**Figure 1.** Upper tail probabilities for f(w) = max(|w1|, |w2|) (left) and g(w) = max(w1, w2) (right) when applied to Y (solid) and X (dotted), where Y and X are centered multivariate normals with ΣY ⪰ ΣX. The plot on the left reflects Anderson’s Theorem (Anderson, 1955), while the plot on the right illustrates Corollary 1. values, g(w) is also quasiconvex and continuous but is not mirror symmetric, resulting in stochastic domi… view at source ↗

**Figure 2.** Upper tail probabilities for f(w) = max{|w1|, |w2|} (left) and g(w) = max{w1, w2} (right) when applied to Y (solid) and X (dotted), where Y and X are centered multivariate normals such that Y has larger marginal variances than X, Y and X satisfy the covariance conditions of the Sudakov-Fernique inequality, but ΣY ⪰̸ ΣX. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

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