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The scaled Gaussian covariance has matrix depth at least one half

Statement

Let $Σ ⪰ 0$ , let $U \subseteq S^{p - 1}$ be nonempty, and let $N_{p} (0, Σ)$ be its centered Gaussian law. If $β$ is the unique positive constant satisfying $Φ (\sqrt{β}) = 3 / 4$ , then $D_{U} (β Σ, N_{p} (0, Σ)) \geq \frac{1}{2} .$ Equality holds for every direction with positive variance; when $Σ = 0$ the depth is $1$ .

Facts & Assumptions

Given: A positive semidefinite covariance matrix $Σ$ , a nonempty direction set $U \subseteq S^{p - 1}$ , and its centered Gaussian law.

[F1]

Matrix depth and covariance estimators defines depth using both weak tails, including their boundary ties.

[F2]

Multivariate normal law, including singular covariance: for $u \in R^{p}$ , $u^{T} X \sim N (0, u^{T} Σ u)$ under $N_{p} (0, Σ)$ .

[F3]

The Gaussian matrix-depth scale constant exists uniquely: $Φ (\sqrt{β}) = 3 / 4$ for the unique $β > 0$ .

[F4]

A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere: a nonnegative random variable with expectation zero vanishes almost surely.

Proof

1.1F2F3algebra

Fix $u \in U$ and put $v = u^{T} Σ u \geq 0$ . If $v > 0$ , [F2] gives $u^{T} X = \sqrt{v} Z$ with $Z \sim N (0, 1)$ . The threshold for $β Σ$ is $u^{T} (β Σ) u = β v$ , so $P ((u^{T} X)^{2} \leq β v) = P (∣ Z ∣ \leq \sqrt{β}) = 2 Φ (\sqrt{β}) - 1 = \frac{1}{2} .$ The normal law has a density, so its boundary has probability zero and $P ((u^{T} X)^{2} \geq β v) = 1 / 2$ as well.

2.1F1F2F4step 1.1algebra∎

If $v = 0$ , then $E (u^{T} X)^{2} = v = 0$ by [F2]. Fact [F4] implies $u^{T} X = 0$ almost surely. Both weak-tail events at threshold zero then have probability one. Thus in every direction the smaller weak tail is at least $1 / 2$ ; taking the infimum proves the depth bound. If $Σ = 0$ , every direction is in the zero-variance case and the depth equals one.

Depends on

Used by

Operator error bound for the Gaussian Huber matrix-depth estimator Lemma

Dependency tree · two levels

42 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model, Proposition 3.1 (standard reference, not scraped)