Testing for Independence of Large Dimensional Vectors

Abstract

In this paper new tests for the independence of two high-dimensional vectors are investigated. We consider the case where the dimension of the vectors increases with the sample size and propose multivariate analysis of variance-type statistics for the hypothesis of a block diagonal covariance matrix. The asymptotic properties of the new test statistics are investigated under the null hypothesis and the alternative hypothesis using random matrix theory. For this purpose we study the weak convergence of linear spectral statistics of central and (conditionally) non-central Fisher matrices. In particular, a central limit theorem for linear spectral statistics of large dimensional (conditionally) non-central Fisher matrices is derived which is then used to analyse the power of the tests under the alternative. The theoretical results are illustrated by means of a simulation study where we also compare the new tests with several alternative, in particular with the commonly used corrected likelihood ratio test. It is demonstrated that the latter test does not keep its nominal level, if the dimension of one sub-vector is relatively small compared to the dimension of the other sub-vector. On the other hand the tests proposed in this paper provide a reasonable approximation of the nominal level in such situations. Moreover, we observe that one of the proposed tests is most powerful under a variety of correlation scenarios.

30 pages main, 21 pages supplement, 14 Figures (accepted for publication in Annals of Statistics)

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• Regularized estimation of large covariance matrices
• Nonlinear shrinkage estimation of large-dimensional covariance matrices
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• High Dimensional Statistical Inference and Random Matrices
• Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy--Widom limits and rates of convergence
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