High-dimensional asymptotic expansions for the distributions of canonical correlations

Abstract

This paper examines asymptotic distributions of the canonical correlations between x 1 ; q × 1 and x 2 ; p × 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 → ∞ and c = p / n → c 0 ∈ [ 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.