High-dimensional asymptotic expansions for the distributions of canonical correlations

This paper examines asymptotic distributions of the canonical correlations between x(1); q x 1 and x(2); p x 1 with q <= p, based on a sample of size of N = n + 1. The asymptotic distributions of the canonical correlations have been studied extensively when the dimensions q and p are fixed and the sample size N tends toward infinity. However, these approximations worsen when q or p is large in comparison to N. To overcome this weakness, this paper first derives asymptotic distributions of the canonical correlations under a high-dimensional framework such that q is fixed, m = n - p --> 00 and c = p/n --> to c(0) is an element of vertical bar 0, 1), assuming that x(1) and x(2) have a joint (q+p)-variate normal distribution. An extended Fisher's z-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, q, and n and the population canonical correlations. (C) 2008 Elsevier Inc. All rights reserved.