Nonlinear shrinkage estimation of large-dimensional covariance matrices

Many statistical applications require an estimate of a covariance matrix and/or its inverse. Whenthe matrix dimension is large compared to the sample size, which happens frequently, the samplecovariance matrix is known to perform poorly and may suffer from ill-conditioning. There alreadyexists an extensive literature concerning improved estimators in such situations. In the absence offurther knowledge about the structure of the true covariance matrix, the most successful approachso far, arguably, has been shrinkage estimation. Shrinking the sample covariance matrix to amultiple of the identity, by taking a weighted average of the two, turns out to be equivalent tolinearly shrinking the sample eigenvalues to their grand mean, while retaining the sampleeigenvectors. Our paper extends this approach by considering nonlinear transformations of thesample eigenvalues. We show how to construct an estimator that is asymptotically equivalent toan oracle estimator suggested in previous work. As demonstrated in extensive Monte Carlosimulations, the resulting bona fide estimator can result in sizeable improvements over the samplecovariance matrix and also over linear shrinkage. DOI: https://doi.org/10.1214/12-AOS989 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-64637 Published Version Originally published at: Ledoit, Olivier; Wolf, Michael (2012). Nonlinear shrinkage estimation of large-dimensional covariance matrices. The Annals of Statistics, 40(2):1024-1060. DOI: https://doi.org/10.1214/12-AOS989 The Annals of Statistics 2012, Vol. 40, No. 2, 1024–1060 DOI: 10.1214/12-AOS989 © Institute of Mathematical Statistics, 2012 NONLINEAR SHRINKAGE ESTIMATION OF LARGE-DIMENSIONAL COVARIANCE MATRICES BY OLIVIER LEDOIT AND MICHAEL WOLF1 University of Zurich 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.