Law of Log Determinant of Sample Covariance Matrix and Optimal Estimation of Differential Entropy for High-Dimensional Gaussian Distributions

• Activity Number: 136
• Type: Contributed
• Date/Time: Monday, August 5, 2013 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Tengyuan Liang*+ and Tony Cai and Harrison Zhou
• Companies: University of Pennsylvania and University of Pennsylvania and Yale University
• Keywords: asymptotic optimality ; covariance matrix ; determinant ; differential entropy ; minimax lower bound ; sharp minimaxity

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 $p$ 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 $p/n \rightarrow 0$ the estimator is asymptotically sharp minimax. The ultra-high dimensional setting where $p > n$ is also discussed. This joint work with Tony Cai and Harrison Zhou.