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June 2016 Nonparametric eigenvalue-regularized precision or covariance matrix estimator
Clifford Lam
Ann. Statist. 44(3): 928-953 (June 2016). DOI: 10.1214/15-AOS1393

Abstract

We introduce nonparametric regularization of the eigenvalues of a sample covariance matrix through splitting of the data (NERCOME), and prove that NERCOME enjoys asymptotic optimal nonlinear shrinkage of eigenvalues with respect to the Frobenius norm. One advantage of NERCOME is its computational speed when the dimension is not too large. We prove that NERCOME is positive definite almost surely, as long as the true covariance matrix is so, even when the dimension is larger than the sample size. With respect to the Stein’s loss function, the inverse of our estimator is asymptotically the optimal precision matrix estimator. Asymptotic efficiency loss is defined through comparison with an ideal estimator, which assumed the knowledge of the true covariance matrix. We show that the asymptotic efficiency loss of NERCOME is almost surely 0 with a suitable split location of the data. We also show that all the aforementioned optimality holds for data with a factor structure. Our method avoids the need to first estimate any unknowns from a factor model, and directly gives the covariance or precision matrix estimator, which can be useful when factor analysis is not the ultimate goal. We compare the performance of our estimators with other methods through extensive simulations and real data analysis.

Citation

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Clifford Lam. "Nonparametric eigenvalue-regularized precision or covariance matrix estimator." Ann. Statist. 44 (3) 928 - 953, June 2016. https://doi.org/10.1214/15-AOS1393

Information

Received: 1 January 2015; Revised: 1 September 2015; Published: June 2016
First available in Project Euclid: 11 April 2016

zbMATH: 1341.62124
MathSciNet: MR3485949
Digital Object Identifier: 10.1214/15-AOS1393

Subjects:
Primary: 62H12
Secondary: 15B52 , 62G20

Keywords: Covariance matrix , data splitting , factor model , High dimensional data analysis , nonlinear shrinkage , Stieltjes transform

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 3 • June 2016
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