Open Access
December 2015 Estimation of functionals of sparse covariance matrices
Jianqing Fan, Philippe Rigollet, Weichen Wang
Ann. Statist. 43(6): 2706-2737 (December 2015). DOI: 10.1214/15-AOS1357


High-dimensional statistical tests often ignore correlations to gain simplicity and stability leading to null distributions that depend on functionals of correlation matrices such as their Frobenius norm and other $\ell_{r}$ norms. Motivated by the computation of critical values of such tests, we investigate the difficulty of estimation the functionals of sparse correlation matrices. Specifically, we show that simple plug-in procedures based on thresholded estimators of correlation matrices are sparsity-adaptive and minimax optimal over a large class of correlation matrices. Akin to previous results on functional estimation, the minimax rates exhibit an elbow phenomenon. Our results are further illustrated in simulated data as well as an empirical study of data arising in financial econometrics.


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Jianqing Fan. Philippe Rigollet. Weichen Wang. "Estimation of functionals of sparse covariance matrices." Ann. Statist. 43 (6) 2706 - 2737, December 2015.


Received: 1 February 2015; Revised: 1 June 2015; Published: December 2015
First available in Project Euclid: 7 October 2015

zbMATH: 1327.62338
MathSciNet: MR3405609
Digital Object Identifier: 10.1214/15-AOS1357

Primary: 62H12
Secondary: 62C20 , 62H15 , 62H25

Keywords: Covariance matrix , elbow effect , Functional estimation , high-dimensional testing , minimax

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 6 • December 2015
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