Test of independence for high-dimensional random vectors based on freeness in block correlation matrices
Date of Issue2017
School of Physical and Mathematical Sciences
In this paper, we are concerned with the independence test for kk high-dimensional sub-vectors of a normal vector, with fixed positive integer kk. 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), 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 kk sub-vectors. Simulated results show the effect of the Schott type statistic, in contrast to those statistics proposed in Jiang and Yang (2013) and Jiang, Bai and Zheng (2013), is satisfactory. Real data analysis is also used to illustrate our method.
Block correlation matrix
Electronic Journal of Statistics
© 2017 The author(s) (published by The Institute of Mathematical Statistics and The Bernoulli Society for Mathematical Statistics and Probability). This work is licensed under a Creative Commons Attribution 4.0 International License.