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

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 ∗Corresponding author. †Z.G. Bao was supported by a startup fund from HKUST. ‡J. Hu was partially supported by Science and Technology Development Foundation of Jilin (Grant No. 20160520174JH), Science and Technology Foundation of Jilin during the “13th Five-Year Plan” and the National Natural Science Foundation of China (Grant No. 11301063). §G.M. Pan was partially supported by a MOE Tier 2 grant 2014-T2-2-060 and by a MOE Tier 1 Grant RG25/14 at the Nanyang Technological University, Singapore. ¶W. Zhou was partially supported by R-155-000-165-112 at the National University of Singapore. 1527

[1]  B. Collins,et al.  Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group , 2004, math-ph/0402073.

[2]  Alexander Soshnikov,et al.  Central limit theorem for traces of large random symmetric matrices with independent matrix elements , 1998 .

[3]  K. Pearson.  On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is Such that it Can be Reasonably Supposed to have Arisen from Random Sampling , 1900 .

[4]  Shu-rong Zheng,et al.  Testing the independence of sets of large-dimensional variables , 2013 .

[5]  Olivier Ledoit,et al.  Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size , 2002 .

[6]  T. Cai,et al.  Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices , 2011, 1102.2925.

[7]  T. W. Anderson.  An Introduction to Multivariate Statistical Analysis, 2nd Edition. , 1985 .

[8]  Z. Bai,et al.  Corrections to LRT on large-dimensional covariance matrix by RMT , 2009, 0902.0552.

[9]  James R. Schott,et al.  Testing for complete independence in high dimensions , 2005 .

[10]  R. Cont.  Empirical properties of asset returns: stylized facts and statistical issues , 2001 .

[11]  Piotr Sniady,et al.  Second order freeness and fluctuations of random matrices: II. Unitary random matrices , 2007 .

[12]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

[13]  Z. Bai,et al.  CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data , 2017, Statistical Papers.

[14]  S. S. Wilks.  On the Independence of k Sets of Normally Distributed Statistical Variables , 1935 .

[15]  N. Reid,et al.  Testing the structure of the covariance matrix with fewer observations than the dimension , 2012, J. Multivar. Anal..

[16]  A. Edelman,et al.  Statistical eigen-inference from large Wishart matrices , 2007, math/0701314.

[17]  D. Voiculescu.  Limit laws for Random matrices and free products , 1991 .

[18]  Yongcheng Qi,et al.  Likelihood Ratio Tests for High-Dimensional Normal Distributions , 2015 .

[19]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[20]  Mariya Shcherbina,et al.  Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices , 2011, 1101.3249.

[21]  K. Johansson.  On fluctuations of eigenvalues of random Hermitian matrices , 1998 .

[22]  L. Pastur,et al.  CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES , 2008, 0809.4698.

[23]  I. Johnstone.  On the distribution of the largest eigenvalue in principal components analysis , 2001 .

[24]  M. Srivastava.  Some Tests Concerning the Covariance Matrix in High Dimensional Data , 2005 .

[25]  Piotr Sniady,et al.  Second order freeness and fluctuations of random matrices. III: Higher order freeness and free cumulants , 2006, Documenta Mathematica.

[26]  Roland Speicher,et al.  Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces , 2004, math/0405191.

[27]  J. Mingo,et al.  Real second order freeness and Haar orthogonal matrices , 2012, 1210.6097.

[28]  Yanrong Yang,et al.  Independence test for high dimensional data based on regularized canonical correlation coefficients , 2015, 1503.05324.

[29]  Real Second-Order Freeness and the Asymptotic Real Second-Order Freeness of Several Real Matrix Models , 2011, 1101.0422.