Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Covariance estimation with direction dependence accuracy

Pedro Abdalla, Shahar Mendelson

Abstract

We construct an estimator $\widehatΣ$ for covariance matrices of unknown, centred random vectors X, with the given data consisting of N independent measurements $X_1,...,X_N$ of X and the wanted confidence level. We show under minimal assumptions on X, the estimator performs with the optimal accuracy with respect to the operator norm. In addition, the estimator is also optimal with respect to direction dependence accuracy: $\langle \widehatΣu,u\rangle$ is an optimal estimator for $σ^2(u)=\mathbb{E}\langle X,u\rangle^2$ when $σ^2(u)$ is ``large".

Covariance estimation with direction dependence accuracy

Abstract

We construct an estimator for covariance matrices of unknown, centred random vectors X, with the given data consisting of N independent measurements of X and the wanted confidence level. We show under minimal assumptions on X, the estimator performs with the optimal accuracy with respect to the operator norm. In addition, the estimator is also optimal with respect to direction dependence accuracy: is an optimal estimator for when is ``large".
Paper Structure (11 sections, 19 theorems, 189 equations)

This paper contains 11 sections, 19 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Lower Bounds
  3. The $L_2$ Isomorphic Distance Oracle
  4. Estimation in Orthogonal Subspaces
  5. Proof of Proposition \ref{['prop:estimation_E_Eperp']} $(1)$: Estimation in $E^{\perp}$
  6. Proof of Proposition \ref{['prop:estimation_E_Eperp']} (2): Estimation in $E$
  7. Estimation in the Euclidean Sphere
  8. Construction of an Admissible Sequence
  9. Estimating the Term II
  10. Estimating the terms $(I)$ and $(III)$:
  11. Proof of Lemma \ref{['lem:application_talagrand_MM']}

Key Result

Theorem 1.6

There exist an absolute constant $\kappa_E$ and a constant $\kappa_{up}(\kappa)$ depending only on $\kappa$ for which the following hold. For every $\delta\in (0,1)$, there exists an estimator $\widehat{\Sigma}(X_1,\ldots,X_N,\delta)$ satisfying, with probability at least $1-\delta$ with respect to

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.6: Main Result
  • Remark 1.7
  • Theorem 1.8: Lower Bound
  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • ...and 26 more