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 ${\ell }_r$ 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.

Published at http://dx.doi.org/10.1214/15-AOS1357 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (12)

• Regularized estimation of large covariance matrices
• Covariance regularization by thresholding
• A two-sample test for high-dimensional data with applications to gene-set testing
• Optimal rates of convergence for covariance matrix estimation
• Operator norm consistent estimation of large-dimensional sparse covariance matrices
• Asymptotic power of sphericity tests for high-dimensional data
• Adaptive covariance matrix estimation through block thresholding
• High-dimensional analysis of semidefinite relaxations for sparse principal components
• Augmented sparse principal component analysis for high dimensional data
• Nonquadratic estimators of a quadratic functional
• Goodness-of-fit testing and quadratic functional estimation from indirect observations
• Optimal adaptive estimation of a quadratic functional

Cited by in corpus (2)

• Bayesian inference for spectral projectors of the covariance matrix
• Testing Simultaneous Diagonalizability