Nestor Parolya: 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 studied under the null hypothesis and the alternative using  random matrix theory. For this purpose we study  the asymptotic properties of linear spectral statistics of  central and non-central  Fisher matrices.

In particular we derive  a central limit theorem for  linear spectral  statistics of  large dimension  non-central Fisher matrices, which can be 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 the commonly used likelihood ratio test. In particular it is demonstrated that the latter test does not keep  its nominal level, if the dimension of one vector is relatively small  compared to the dimension of the other vector. On the other hand the tests proposed in this paper provide a reasonable approximation of the nominal level in such situations.