A hypothesis test for independence of sets of variates in high dimensions

Srivastava [Srivastava, M.S., 2005. Some tests concerning the covariance matrix in high dimensional data. J. Japan Statist. Soc. 35, 251-272] has proposed a test statistic for testing the hypothesis that the covariance matrix of the normal population is a diagonal matrix when the sample size is smaller than the dimensionality of the data. We extend his results to the hypothesis testing problem for independence of sets of variates in high dimensions. A test statistic is proposed and its asymptotic null distribution is also given, as both the sample size and the number of variables go to infinity. Consequently, this test can be used when the number of variables is not small relative to the sample size, in particular, even when the number of variables exceeds the sample size. Simulations are performed to see the accuracy of the asymptotic null distribution.