Statistics Papers

Document Type

Journal Article

Date of this Version

3-2016

Publication Source

The Annals of Statistics

Volume

44

Issue

2

Start Page

455

Last Page

488

DOI

10.1214/13-AOS1171

Abstract

Precision matrix is of significant importance in a wide range of applications in multivariate analysis. This paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting. Optimal rates of convergence are established for a range of matrix norm losses. A fully data driven estimator based on adaptive constrained ℓ1 minimization is proposed and its rate of convergence is obtained over a collection of parameter spaces. The estimator, called ACLIME, is easy to implement and performs well numerically.

A major step in establishing the minimax rate of convergence is the derivation of a rate-sharp lower bound. A “two-directional” lower bound technique is applied to obtain the minimax lower bound. The upper and lower bounds together yield the optimal rates of convergence for sparse precision matrix estimation and show that the ACLIME estimator is adaptively minimax rate optimal for a collection of parameter spaces and a range of matrix norm losses simultaneously.

Copyright/Permission Statement

The original and published work is available at: https://projecteuclid.org/euclid.aos/1458245724#abstract

Keywords

Constrained ℓ1-minimization, covariance matrix, graphical model, minimax lower bound, optimal rate of convergence, precision matrix, sparsity, spectral norm.

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Date Posted: 27 November 2017

This document has been peer reviewed.