Gaussian comparison above the median

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.

Disclosure

“Generative AI and AI-assisted technologies During the preparation of this work, the author used OpenAI’s ChatGPT to help for- mulate the numerical example in §3 and the numerical counterexample to the conjecture of Cohen and Fogarty (2022, §9.1). It was also used to check algebraic and proof details for the author’s proofs of Theorem 1 and Proposi”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-24T00:34:15.228Z

Structural counts