文章基本信息

标题：Test of independence for high-dimensional random vectors based on freeness in block correlation matrices
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作者：Zhigang Bao ; Jiang Hu ; Guangming Pan 等
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2017
卷号：11
期号：1
页码：1527-1548
DOI：10.1214/17-EJS1259
语种：English
出版社：Institute of Mathematical Statistics
摘要：In this paper, we are concerned with the independence test for $k$ high-dimensional sub-vectors of a normal vector, with fixed positive integer $k$. A natural high-dimensional extension of the classical sample correlation matrix, namely block correlation matrix, is proposed for this purpose. We then construct the so-called Schott type statistic as our test statistic, which turns out to be a particular linear spectral statistic of the block correlation matrix. Interestingly, the limiting behavior of the Schott type statistic can be figured out with the aid of the Free Probability Theory and the Random Matrix Theory. Specifically, we will bring the so-called real second order freeness for Haar distributed orthogonal matrices, derived in Mingo and Popa (2013)[10], into the framework of this high-dimensional testing problem. Our test does not require the sample size to be larger than the total or any partial sum of the dimensions of the $k$ sub-vectors. Simulated results show the effect of the Schott type statistic, in contrast to those statistics proposed in Jiang and Yang (2013)[8] and Jiang, Bai and Zheng (2013)[7], is satisfactory. Real data analysis is also used to illustrate our method.