Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for high-dimensional Gaussian distributions
Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for high-dimensional Gaussian distributions
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DOI：
10.1016/j.jmva.2015.02.003
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发表时间：
2015-05-01
影响因子：
1.6
通讯作者：
Zhou, Harrison H.
中科院分区：
文献类型：
--
作者：
Cai, T. Tony;Liang, Tengyuan;Zhou, Harrison H.
多元高斯分布协方差矩阵的微分熵和对数行列式在编码、通信、信号处理和统计推断中有着广泛的应用。本文在高维情形下考虑了协方差矩阵的最优微分熵估计和对数行列式估计。我们首先建立了样本协方差矩阵的对数行列式在高维情形下的中心极限定理，在高维情形下，p(N)可以随着样本容量n的增加而增长，然后考虑微分熵和对数行列式的估计量。得到了最优收敛速度。证明了在p(N)/n->；0的情形下，估计是渐近尖锐的极大极小。还讨论了p(N)>；n的超高维设置。(C)2015 Elsevier Inc.保留所有权利。
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.