# Nonlinear shrinkage estimation of large-dimensional covariance matrices

### Abstract

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.

Annals of
Statistics, Volume 40, Number 2,
April 2012,
pages 1024-1060

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