Statistics Papers

Document Type

Journal Article

Date of this Version

2012

Publication Source

Annals of Statistics

Volume

40

Issue

4

Start Page

2014

Last Page

2042

DOI

10.1214/12-AOS999

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.

Keywords

adaptive estimation, block thresholding, covariance matrix, Frobenius norm, minimax estimation, optimal rate of convergence, spectral norm

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Date Posted: 27 November 2017

This document has been peer reviewed.