High-dimensional asymptotic expansions for the distributions of canonical correlations

• Published in: Journal of multivariate analysis 2009, Vol.100 (1), p.231-242
• Main Authors: Fujikoshi, Yasunori, Sakurai, Tetsuro
• Format: Article
• Language: English
• Subjects: 62E20; 62E20 Asymptotic distributions Canonical correlations Extended Fisher's z-transformation High-dimensional framework; 62H10; 62H10 secondary; Approximation; Asymptotic distributions; Asymptotic methods; Canonical correlations; Distribution theory; Exact sciences and technology; Extended Fisher’s [formula omitted]-transformation; High-dimensional framework; Mathematics; Multivariate analysis; Normal distribution; primary; Probability and statistics; Probability theory and stochastic processes; Random variables; Sample size; Sampling theory, sample surveys; Sciences and techniques of general use; secondary; Statistics; Studies
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2008.04.009

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.