The Annals of Statistics

Nonparametric eigenvalue-regularized precision or covariance matrix estimator

Clifford Lam

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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.

Article information

Ann. Statist., Volume 44, Number 3 (2016), 928-953.

Received: January 2015
Revised: September 2015
First available in Project Euclid: 11 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 62G20: Asymptotic properties 15B52: Random matrices

High dimensional data analysis covariance matrix Stieltjes transform data splitting nonlinear shrinkage factor model


Lam, Clifford. Nonparametric eigenvalue-regularized precision or covariance matrix estimator. Ann. Statist. 44 (2016), no. 3, 928--953. doi:10.1214/15-AOS1393.

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Supplemental materials

  • Simulations and proofs of theorems in the paper. We present five profiles of simulations and compare the performance of NERCOME to other state-of-the-art methods. We also present the proofs of Theorem 1, Theorem 3, Lemma 1, Theorem 5 and Theorem 6 in the paper.