On the Strong Convergence of the Optimal Linear Shrinkage Estimator for Large Dimensional Covariance Matrix

TL;DR

This paper develops an optimal linear shrinkage estimator for large-dimensional covariance matrices, leveraging random matrix theory to achieve almost sure convergence and minimal Frobenius loss in high-dimensional settings.

Contribution

It introduces a distribution-free, asymptotically optimal shrinkage estimator for covariance matrices when both dimension and sample size grow large.

Findings

Estimator achieves almost sure convergence to the true covariance matrix.

The Frobenius norm of the sample covariance matrix converges to a deterministic limit.

The method provides consistent estimation of the optimal shrinkage intensities.

Abstract

In this work we construct an optimal linear shrinkage estimator for the covariance matrix in high dimensions. The recent results from the random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The developed distribution-free estimators obey almost surely the smallest Frobenius loss over all linear shrinkage estimators for the covariance matrix. The case we consider includes the number of variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ so that $p/n\rightarrow c\in (0, +\infty)$. Additionally, we prove that the Frobenius norm of the sample covariance matrix tends almost surely to a deterministic quantity which can be consistently estimated.