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Regularized estimation of large covariance matrices. (English) Zbl 1132.62040
Summary: This paper considers estimating a covariance matrix of \(p\) variables from \(n\) observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse of the covariance. We show that these estimates are consistent in the operator norm as long as \((\log p)/n\rightarrow 0\), and obtain explicit rates. The results are uniform over some fairly natural well-conditioned families of covariance matrices. We also introduce an analogue of the Gaussian white noise model and show that if the population covariance is embeddable in that model and well-conditioned, then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix. The results can be extended to smooth versions of banding and to non-Gaussian distributions with sufficiently short tails. A resampling approach is proposed for choosing the banding parameter in practice. This approach is illustrated numerically on both simulated and real data.

62H12 Estimation in multivariate analysis
62F12 Asymptotic properties of parametric estimators
15A18 Eigenvalues, singular values, and eigenvectors
62F40 Bootstrap, jackknife and other resampling methods
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