Testing for independence of high-dimensional variables: [formula omitted]-coefficient based approach

Abstract

We treat the problem of testing mutual independence of k high-dimensional random vectors when the data are multivariate normal and k≥2 is a fixed integer. For this purpose, we focus on the vector correlation coefficient, ρV and propose an extension of its classical estimator which is constructed to correct potential sources of inconsistency related to the high dimensionality. Building on the proposed estimator of ρV, we derive the new test statistic and study its limiting behavior in a general high-dimensional asymptotic framework which allows the vector’s dimensionality arbitrarily exceed the sample size. Specifically, we show that the asymptotic distribution of the test statistic under the main hypothesis of independence is standard normal and that the proposed test is size and power consistent. Using our statistics, we further construct the step-down multiple comparison procedure based on the closed testing strategy for the simultaneous test for independence. Accuracy of the proposed tests in finite samples is shown through simulations for a variety of high-dimensional scenarios in combination with a number of alternative dependence structures. Real data analysis is performed to illustrate the utility of the test procedures.