Test of Independence for High-dimensional Random Vectors Based on Block Correlation Matrices
Abstract
In this paper, we are concerned with the independence test for high-dimensional sub-vectors of a normal vector, with fixed positive integer . A natural high-dimensional extension of the classical sample correlation matrix, namely block correlation matrix, is raised 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 \cite{MP2013}, 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 sub-vectors. Simulated results show the effect of the Schott type statistic, in contrast to those of the statistics proposed in \cite{JY2013} and \cite{JBZ2013}, is satisfactory. Real data analysis is also used to illustrate our method.
Cite
@article{arxiv.1410.5082,
title = {Test of Independence for High-dimensional Random Vectors Based on Block Correlation Matrices},
author = {Zhigang Bao and Jiang Hu and Guangming Pan and Wang Zhou},
journal= {arXiv preprint arXiv:1410.5082},
year = {2014}
}