Test of Independence for High-dimensional Random Vectors Based on Block Correlation Matrices

TL;DR

This paper introduces a new high-dimensional independence test based on block correlation matrices and a Schott type statistic, utilizing Free Probability and Random Matrix Theory for its theoretical foundation.

Contribution

It proposes a novel independence testing method for high-dimensional vectors that does not require large sample sizes, extending classical correlation concepts with advanced spectral analysis.

Findings

The Schott type statistic effectively tests independence in high dimensions.

The method's theoretical properties are derived using Free Probability and Random Matrix Theory.

Simulations and real data demonstrate the test's satisfactory performance.

Abstract

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 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 $k$ 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.