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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.

MSC:
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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