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 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.
Highlights
Test of independence for random variables is a very classical hypothesis testing problem, which dates back to the seminal work by Pearson in [17], followed by a huge literature regarding this topic and its variants
One frequently recurring variant is the test of independence for k random vectors, where k ≥ 2 is an integer
Our aim in this paper is to propose a new statistic with both statistical visualizability and mathematical tractability, whose limiting behavior can be derived with the following restriction on the dimensionality and the sample size pi := pi(n), pi/n → yi ∈ (0, 1), i ∈ JkK, as n → ∞
Summary
Test of independence for random variables is a very classical hypothesis testing problem, which dates back to the seminal work by Pearson in [17], followed by a huge literature regarding this topic and its variants. To employ the existing RMT apparatus in the sequel, hereafter, we will always work with the above large n and proportionally large p setting This time, resorting to the likelihood ratio statistic in (1.2) directly is obviously infeasible, since the limiting law (1.2) is invalid when p tends to infinity along with n. Our aim in this paper is to propose a new statistic with both statistical visualizability and mathematical tractability, whose limiting behavior can be derived with the following restriction on the dimensionality and the sample size pi := pi(n), pi/n → yi ∈ (0, 1), i ∈ JkK, as n → ∞. Employing this limiting law to test independence of k sub-vectors under (1.5) and assessing the statistic via simulations and a real data set, which comes from the daily returns of 258 stocks issued by the companies from S&P 500. For random objects ξ and η, we use ξ =d η to represent that ξ and η share the same distribution
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