title: ESTIMATION OF FUNCTIONALS OF SPARSE COVARIANCE MATRICES.
abstract: 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 ℓrnorms. 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.
authors.affiliation: Department of Operations Research and Financial Engineering Princeton University Princeton, New Jersey 08544 USA.
authors.forename: Jianqing
authors.initials: J
authors.lastname: Fan
authors.affiliation: Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, Massachusetts 02139-4307 USA rigollet@math.mit.edu.
authors.forename: Philippe
authors.initials: P
authors.lastname: Rigollet
authors.affiliation: Department of Operations Research and Financial Engineering Princeton University Princeton, New Jersey 08544 USA.
authors.forename: Weichen
authors.initials: W
authors.lastname: Wang
keywords: Covariance matrix; elbow effect; functional estimation; high-dimensional testing; minimax