Adaptive covariance matrix estimation through block thresholding

TL;DR

This paper introduces a data-driven block thresholding method for adaptive covariance matrix estimation that achieves minimax rate optimality across various parameter spaces, with strong theoretical guarantees and promising simulation results.

Contribution

It proposes a novel fully data-driven block thresholding estimator that is adaptively minimax rate optimal over a wide class of covariance matrices.

Findings

Estimator is minimax rate optimal adaptively.

Performs well in simulations.

Effective across diverse covariance structures.

Abstract

Estimation of large covariance matrices has drawn considerable recent attention, and the theoretical focus so far has mainly been on developing a minimax theory over a fixed parameter space. In this paper, we consider adaptive covariance matrix estimation where the goal is to construct a single procedure which is minimax rate optimal simultaneously over each parameter space in a large collection. A fully data-driven block thresholding estimator is proposed. The estimator is constructed by carefully dividing the sample covariance matrix into blocks and then simultaneously estimating the entries in a block by thresholding. The estimator is shown to be optimally rate adaptive over a wide range of bandable covariance matrices. A simulation study is carried out and shows that the block thresholding estimator performs well numerically. Some of the technical tools developed in this paper can also be of independent interest.