Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices

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 happensfrequently, the sample covariance matrix is known to perform poorly and may suffer fromill-conditioning. There already exists an extensive literature concerning improved estimatorsin such situations. In the absence of further knowledge about the structure of thetrue covariance matrix, the most successful approach so far, arguably, has been shrinkageestimation. Shrinking the sample covariance matrix to a multiple of the identity, by takinga weighted average of the two, turns out to be equivalent to linearly shrinking the sampleeigenvalues to their grand mean, while retaining the sample eigenvectors. Our paperextends this approach by considering nonlinear transformations of the sample eigenvalues.We show how to construct an estimator that is asymptotically equivalent to an oracleestimator suggested in previous work. As demonstrated in extensive Monte Carlo simulations,the resulting bona fide estimator can result in sizeable improvements over the samplecovariance matrix and also over linear shrinkage.[...]