Nonlinear shrinkage estimation of large-dimensional covariance matrices

TL;DR

This paper introduces a nonlinear shrinkage method for estimating large-dimensional covariance matrices, improving accuracy over traditional linear methods especially in high-dimensional, small-sample scenarios.

Contribution

It extends linear shrinkage estimators by developing a nonlinear eigenvalue transformation approach, achieving asymptotic optimality and better empirical performance.

Findings

Significant improvement over sample covariance matrices in simulations

Nonlinear shrinkage outperforms linear methods in accuracy

Estimator is asymptotically equivalent to an oracle estimator

Abstract

Many statistical applications require an estimate of a covariance matrix and/or its inverse. When the matrix dimension is large compared to the sample size, which happens frequently, the sample covariance matrix is known to perform poorly and may suffer from ill-conditioning. There already exists an extensive literature concerning improved estimators in such situations. In the absence of further knowledge about the structure of the true covariance matrix, the most successful approach so far, arguably, has been shrinkage estimation. Shrinking the sample covariance matrix to a multiple of the identity, by taking a weighted average of the two, turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean, while retaining the sample eigenvectors. Our paper extends this approach by considering nonlinear transformations of the sample eigenvalues. We show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work. As demonstrated in extensive Monte Carlo simulations, the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage.