Shrinkage estimation of large covariance matrices: Keep it simple, statistician?

• Published in: Journal of multivariate analysis Vol. 186; p. 104796
• Main Authors: Ledoit, Olivier, Wolf, Michael
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.11.2021
• Subjects: Large-dimensional asymptotics; Random matrix theory; Rotation equivariance; secondary; Rotation equivariance; Large-dimensional asymptotics; Random matrix theory; primary
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2021.104796

Under rotation-equivariant decision theory, sample covariance matrix eigenvalues can be optimally shrunk by recombining sample eigenvectors with a (potentially nonlinear) function of the unobservable population covariance matrix. The optimal shape of this function reflects the loss/risk that is to be minimized. We solve the problem of optimal covariance matrix estimation under a variety of loss functions motivated by statistical precedent, probability theory, and differential geometry. A key ingredient of our nonlinear shrinkage methodology is a new estimator of the angle between sample and population eigenvectors, without making strong assumptions on the population eigenvalues. We also introduce a broad family of covariance matrix estimators that can handle all regular functional transformations of the population covariance matrix under large-dimensional asymptotics. In addition, we compare via Monte Carlo simulations our methodology to two simpler ones from the literature, linear shrinkage and shrinkage based on the spiked covariance model.