NONLINEAR SHRINKAGE ESTIMATION OF LARGE-DIMENSIONAL COVARIANCE MATRICES

• 成果类型: Article
• 署名作者: Ledoit, Olivier; Wolf, Michael
• 署名单位: University of Zurich
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS989
• 发表日期: 2012
• 页码: 1024-1060
• 关键词: limiting spectral distribution; eigenvalues

摘要

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.