## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 11, Number 1 (2017), 1527-1548.

### Test of independence for high-dimensional random vectors based on freeness in block correlation matrices

Zhigang Bao, Jiang Hu, Guangming Pan, and Wang Zhou

#### 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.

#### Article information

**Source**

Electron. J. Statist., Volume 11, Number 1 (2017), 1527-1548.

**Dates**

Received: September 2016

First available in Project Euclid: 19 April 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1492588988

**Digital Object Identifier**

doi:10.1214/17-EJS1259

**Mathematical Reviews number (MathSciNet)**

MR3635921

**Zentralblatt MATH identifier**

1362.62123

**Subjects**

Primary: 62H15: Hypothesis testing

Secondary: 46L54: Free probability and free operator algebras

**Keywords**

Block correlation matrix independence test high dimensional data Schott type statistic second order freeness haar distributed orthogonal matrices central limit theorem random matrices

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Bao, Zhigang; Hu, Jiang; Pan, Guangming; Zhou, Wang. Test of independence for high-dimensional random vectors based on freeness in block correlation matrices. Electron. J. Statist. 11 (2017), no. 1, 1527--1548. doi:10.1214/17-EJS1259. https://projecteuclid.org/euclid.ejs/1492588988