Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for high-dimensional Gaussian distributions

JOURNAL OF MULTIVARIATE ANALYSIS

卷:137

Article

Cai, T. Tony¹  Liang, Tengyuan¹  Zhou, Harrison H.² 

[1] Univ Penn, Wharton Sch, Dept Stat, Philadelphia, PA 19104 USA

[2] Yale Univ, Dept Stat, New Haven, CT 06511 USA

关键词: Asymptotic optimality;    Central limit theorem;    Covariance matrix;    Determinant;    Differential entropy;    Minimax lower bound;    Sharp minimaxity;   

DOI  :  10.1016/j.jmva.2015.02.003

来源: Elsevier

【 摘 要 】

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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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