Asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis when the dimension is large

This paper examines asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis based on a sample of size N=n+1N=n+1 on two sets of variables, i.e., xu;p1×1 and xv;p2×1. These problems are related to dimension reduction. The asymptotic approximations of the statistics have been studied extensively when dimensions p1p1 and p2p2 are fixed and the sample size NN tends to infinity. However, the approximations worsen as p1p1 and p2p2 increase. This paper derives asymptotic expansions of the test statistics when both the sample size and dimension are large, assuming that xu and xv have a joint (p1+p2)(p1+p2)-variate normal distribution. Numerical simulations revealed that this approximation is more accurate than the classical approximation as the dimension increases.