CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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A bounded covariance class forces a capped lower rate

Statement refuted

For fixed $M > 0$ , there are positive constants $C_{M}, c_{0}$ such that for every $p, n \geq 1$ and $0 \leq ϵ \leq 1$ , $\inf_{\hat{Σ}} \sup_{Σ \in F_{p} (M)} \sup_{Q} P_{ϵ, Σ, Q}^{\otimes n} {∥ \hat{Σ} - Σ ∥_{o p}^{2} \geq C_{M} ((p / n) \lor ϵ^{2})} \geq c_{0},$ where the infimum ranges over all Borel matrix-valued estimators.

Facts & Assumptions

Given: AC, a fixed $M > 0$ , proposed positive constants $C_{M}, c_{0}$ , and the bounded-spectrum Gaussian Huber model.

[F1]

Bounded-spectrum covariance class: every $Σ \in F_{p} (M)$ is positive semidefinite with $∥ Σ ∥_{o p} \leq M$ .

[F2]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: changing the sign of a matrix preserves its operator norm.

[F3]

Every complete ordered field is Archimedean: the natural numbers are unbounded above in the real field.

[F4]

Gaussian Huber contamination model for covariance estimation gives the Gaussian Huber sample laws. Their Gaussian convention retains The Axiom of Choice; the pointwise loss bound below is independent of the sampling law.

Counterexample

1.1F1F2F4givenalgebra

The constant zero-matrix estimator is Borel, since the inverse image of any Borel set is either the entire sample space or empty. For every target in the class its squared operator loss is $∥ 0 - Σ ∥_{o p}^{2} = ∥ Σ ∥_{o p}^{2} \leq M^{2}$ by [F1]–[F2], for every possible observation and every contaminant.

2.1F3F4step 1.1givenchoosealgebra∎

By [F3], choose an integer $p > M^{2} / C_{M}$ , and take $n = 1$ , $ϵ = 0$ . The proposed threshold is $C_{M} p > M^{2}$ , so the tail event for the zero estimator is empty under every target and contaminant by step 1.1. Its worst-case probability, hence the nonnegative minimax infimum, is zero, contradicting $c_{0} > 0$ . Thus an uncapped all-dimension lower rate is false; the bounded loss requires a cap such as $\min {1, (p / n) \lor ϵ^{2}}$ in a valid matching lower statement.

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