High-dimensional asymptotic expansions for the distributions of canonical correlations

This paper examines asymptotic distributions of the canonical correlations between x1;q×1 and x2;p×1 with q≤pq≤p, based on a sample of size of N=n+1N=n+1. The asymptotic distributions of the canonical correlations have been studied extensively when the dimensions qq and pp are fixed and the sample size NN tends toward infinity. However, these approximations worsen when qq or pp is large in comparison to NN. To overcome this weakness, this paper first derives asymptotic distributions of the canonical correlations under a high-dimensional framework such that qq is fixed, m=n−p→∞m=n−p→∞ and c=p/n→c0∈[0,1)c=p/n→c0∈[0,1), assuming that x1 and x2 have a joint (q+p)(q+p)-variate normal distribution. An extended Fisher’s zz-transformation is proposed. Then, the asymptotic distributions are improved further by deriving their asymptotic expansions. Numerical simulations revealed that our approximations are more accurate than the classical approximations for a large range of p,qp,q, and nn and the population canonical correlations.