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The Gaussian Huber covariance mixture is a probability law

Statement

For every admissible $p, ϵ, Σ, Q$ in Gaussian Huber contamination model for covariance estimation, $P_{ϵ, Σ, Q}$ is a probability measure on the Borel subsets of $R^{p}$ .

Facts & Assumptions

Given: A probability law $N_{p} (0, Σ)$ , a Borel probability law $Q$ , and $0 \leq ϵ \leq 1$ .

[F1]

Gaussian Huber contamination model for covariance estimation defines $P_{ϵ, Σ, Q} = (1 - ϵ) N_{p} (0, Σ) + ϵ Q$ .

[F2]

Probability measures and probability spaces: both component laws are nonnegative, countably additive, and have total mass one.

Proof

1.1F1F2givenalgebra

For each Borel set $A$ , the mixture assigns $P_{ϵ, Σ, Q} (A) = (1 - ϵ) N_{p} (0, Σ) (A) + ϵ Q (A)$ , which is nonnegative because both coefficients are nonnegative. Its value at the empty set is zero because each component measure vanishes there.

2.1F1F2step 1.1algebra∎

Let $(A_{j})_{j \geq 1}$ be pairwise disjoint Borel sets. Countable additivity of the two component laws in [F2] gives their values on the union as the corresponding nonnegative series; distributing the fixed nonnegative coefficients over partial sums and taking limits yields $P_{ϵ, Σ, Q} (⋃_{j} A_{j}) = \sum_{j} P_{ϵ, Σ, Q} (A_{j})$ . The total mass is $(1 - ϵ) + ϵ = 1$ . Together with step 1.1, these are the probability-measure axioms.

Depends on

Used by

A covariance matrix with one correlated block Example
Dense Gaussian Huber covariance model Example

Cited to discharge well-definedness by Gaussian Huber contamination model for covariance estimation.

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Sources

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model, §2 (standard reference, not scraped)