Open Access
October 2019 Testing for independence of large dimensional vectors
Taras Bodnar, Holger Dette, Nestor Parolya
Ann. Statist. 47(5): 2977-3008 (October 2019). DOI: 10.1214/18-AOS1771


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) noncentral Fisher matrices. In particular, a central limit theorem for linear spectral statistics of large dimensional (conditionally) noncentral 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.


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Taras Bodnar. Holger Dette. Nestor Parolya. "Testing for independence of large dimensional vectors." Ann. Statist. 47 (5) 2977 - 3008, October 2019.


Received: 1 August 2017; Revised: 1 May 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114935
MathSciNet: MR3988779
Digital Object Identifier: 10.1214/18-AOS1771

Primary: 60B20 , 60F05 , 62H15
Secondary: 62F05 , 62H20

Keywords: asymptotic normality , large dimensional covariance matrix , Linear spectral statistics , noncentral Fisher random matrix , Testing for independence

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 5 • October 2019
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