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

Ledoit, Olivier; Wolf, Michael (2021). Shrinkage estimation of large covariance matrices: keep it simple, statistician? Working paper series / Department of Economics 327, University of Zurich.

## Abstract

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.

## Abstract

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.

## Statistics

Detailed statistics

Item Type: Working Paper 03 Faculty of Economics > Department of Economics Working Paper Series > Department of Economics 330 Economics C13 Large-dimensional asymptotics, random matrix theory, rotation equivariance, Kovarianzfunktion, Risikomanagement, Verlust, Modellierung, Eigenwert, Monte-Carlo-Simulation Discipline-based scholarship (basic research) English June 2021 17 Jul 2019 14:03 06 Mar 2024 14:30 Working paper series / Department of Economics 34 1664-705X Revised version Green merlin-id:18370