We construct an estimator for covariance matrices of unknown, centred random vectors X, with the given data consisting of N independent measurements <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X_1,\ldots ,X_N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>N</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> of X and the wanted confidence level. We show under minimal assumptions on X, the estimator <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\widehat{\Sigma }$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi mathvariant="normal">Σ</mi> <mo stretchy="true">^</mo> </mover> </math></EquationSource> </InlineEquation> performs with the optimal accuracy with respect to the operator norm. In addition, the estimator is also optimal with respect to direction dependence accuracy: <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\langle \widehat{\Sigma }u,u\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mover accent="true"> <mi mathvariant="normal">Σ</mi> <mo stretchy="true">^</mo> </mover> <mi>u</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> is an optimal estimator for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sigma ^2(u)=\mathbb {E}\langle X,u\rangle ^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>σ</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="double-struck">E</mi> <msup> <mrow> <mo stretchy="false">⟨</mo> <mi>X</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">⟩</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sigma ^2(u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>σ</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is “large”.

Covariance estimation with direction dependence accuracy

期刊论文
发表日期：

2025年5月2日

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Pedro Abdalla,
Shahar Mendelson

摘要

We construct an estimator for covariance matrices of unknown, centred random vectors X, with the given data consisting of N independent measurements $X_1,\ldots ,X_N$ $X_{1}, \dots, X_{N}$ of X and the wanted confidence level. We show under minimal assumptions on X, the estimator $\widehat{\Sigma }$ $\hat{Σ}$ performs with the optimal accuracy with respect to the operator norm. In addition, the estimator is also optimal with respect to direction dependence accuracy: $\langle \widehat{\Sigma }u,u\rangle $ $⟨ \hat{Σ} u, u ⟩$ is an optimal estimator for $\sigma ^2(u)=\mathbb {E}\langle X,u\rangle ^2$ $σ^{2} (u) = E {⟨ X, u ⟩}^{2}$ when $\sigma ^2(u)$ $σ^{2} (u)$ is “large”.