Testing the independence of sets of large-dimensional variables

This paper proposes the corrected likelihood ratio test (LRT) and large-dimensional trace criterion to test the independence of two large sets of multivariate variables of dimensions p1 and p2 when the dimensions p = p1 + p2 and the sample size n tend to infinity simultaneously and proportionally. Both theoretical and simulation results demonstrate that the traditional χ2 approximation of the LRT performs poorly when the dimension p is large relative to the sample size n, while the corrected LRT and large-dimensional trace criterion behave well when the dimension is either small or large relative to the sample size. Moreover, the trace criterion can be used in the case of p > n, while the corrected LRT is unfeasible due to the loss of definition.