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
Recommended Citation
Cai, T., Liang, T., & Zhou, H. H. (2015). Law of Log Determinant of Sample Covariance Matrix and Optimal Estimation of Differential Entropy for High-Dimensional Gaussian Distributions. Journal of Multivariate Analysis, 137 161-172. http://dx.doi.org/10.1016/j.jmva.2015.02.003
Date Posted: 25 October 2018
This document has been peer reviewed.