The Annals of Statistics

Independence test for high dimensional data based on regularized canonical correlation coefficients

Yanrong Yang and Guangming Pan

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This paper proposes a new statistic to test independence between two high dimensional random vectors $\mathbf{X}:p_{1}\times1$ and $\mathbf{Y}:p_{2}\times1$. The proposed statistic is based on the sum of regularized sample canonical correlation coefficients of $\mathbf{X}$ and $\mathbf{Y}$. The asymptotic distribution of the statistic under the null hypothesis is established as a corollary of general central limit theorems (CLT) for the linear statistics of classical and regularized sample canonical correlation coefficients when $p_{1}$ and $p_{2}$ are both comparable to the sample size $n$. As applications of the developed independence test, various types of dependent structures, such as factor models, ARCH models and a general uncorrelated but dependent case, etc., are investigated by simulations. As an empirical application, cross-sectional dependence of daily stock returns of companies between different sections in the New York Stock Exchange (NYSE) is detected by the proposed test.

Article information

Ann. Statist., Volume 43, Number 2 (2015), 467-500.

First available in Project Euclid: 24 February 2015

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Canonical correlation coefficients central limit theorem large dimensional random matrix theory independence test linear spectral statistics


Yang, Yanrong; Pan, Guangming. Independence test for high dimensional data based on regularized canonical correlation coefficients. Ann. Statist. 43 (2015), no. 2, 467--500. doi:10.1214/14-AOS1284.

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Supplemental materials

  • Supplementary material: Supplement to “Independence test for high dimensional data based on regularized canonical correlation coefficients”. The supplementary material is divided into Appendices A and B. Some useful lemmas, and proofs of all theorems and Proposition 4–5 are given in Appendix A while one theorem related to CLT of a sample covariance matrix plus a perturbation matrix is provided in Appendix B.