Statistics Papers

Document Type

Technical Report

Date of this Version

5-2015

Publication Source

Journal of Multivariate Analysis

Volume

137

Start Page

161

Last Page

172

DOI

10.1016/j.jmva.2015.02.003

Abstract

Differential entropy and log determinant of the covariance matrix of a multivariate Gaussian distribution have many applications in coding, communications, signal processing and statistical inference. In this paper we consider in the high-dimensional setting optimal estimation of the differential entropy and the log-determinant of the covariance matrix. We first establish a central limit theorem for the log determinant of the sample covariance matrix in the high-dimensional setting where the dimension p(n) can grow with the sample size n. An estimator of the differential entropy and the log determinant is then considered. Optimal rate of convergence is obtained. It is shown that in the case p(n) / n → 0 the estimator is asymptotically sharp minimax. The ultra-high-dimensional setting where p(n) > n is also discussed.

Copyright/Permission Statement

© 2015. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

Keywords

asymptotic optimality, central limit theorem, covariance matrix, determinant, differential entropy, minimax lower bound, sharp minimaxity

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Date Posted: 25 October 2018

This document has been peer reviewed.