High-dimensional asymptotic expansions for the distributions of canonical correlations

  • Authors:

  • Yasunori Fujikoshi;Tetsuro Sakurai

  • Affiliations:

  • Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, 112-8551, Japan;Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, 112-8551, Japan

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2009

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Abstract

This paper examines asymptotic distributions of the canonical correlations between x"1;qx1 and x"2;px1 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.