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Nonparametric eigenvalue-regularized precision or covariance matrix estimator

Lam, Clifford (2016) Nonparametric eigenvalue-regularized precision or covariance matrix estimator. Annals of Statistics, 44 (3). pp. 928-953. ISSN 0090-5364

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Identification Number: 10.1214/15-AOS1393


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.

Item Type: Article
Official URL:
Additional Information: © 2015 Institute of Mathematical Statistics
Divisions: Statistics
Subjects: Q Science > QA Mathematics
Date Deposited: 24 Sep 2015 08:28
Last Modified: 20 Oct 2021 02:21

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