Estimation of functionals of sparse covariance matrices

TL;DR

This paper studies the challenge of estimating functionals of sparse correlation matrices in high-dimensional settings, demonstrating that thresholded estimators are optimal and revealing an elbow phenomenon in minimax rates.

Contribution

It proves the minimax optimality of thresholded estimators for functionals of sparse correlation matrices and characterizes the rates with an elbow phenomenon.

Findings

Thresholded estimators are sparsity-adaptive and minimax optimal.

Minimax rates exhibit an elbow phenomenon.

Empirical validation in financial econometrics data.

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.