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.Comment: Published in at http://dx.doi.org/10.1214/12-AOS989 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
