Shrinkage estimation of large covariance matrices: Keep it simple, statistician?
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 b...
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Published in | Journal of multivariate analysis Vol. 186; p. 104796 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Elsevier Inc
01.11.2021
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Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 0047-259X 1095-7243 |
DOI: | 10.1016/j.jmva.2021.104796 |