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

Ledoit, Olivier; Wolf, Michael (2020). 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 English February 2020 17 Jul 2019 14:03 18 Feb 2020 14:37 Working paper series / Department of Economics 48 1664-705X Revised version Green https://www.econ.uzh.ch/static/release/workingpapers.php?id=1006

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