Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations

Muirhead & Waternaux (1980) — canonical correlations without normality

Result

for populations with finite fourth moments, the sample canonical correlations and their test statistics are asymptotically normal with a covariance involving fourth-order cumulants; elliptical simplification and a corrected chi-squared test for zero coefficients.

Use

O7's fourth-moment hypothesis and mechanism; $\eta_D$ is a smooth symmetric function of their $\rho_i^2$, so the vector CLT off the endpoints follows by one delta step. One block of O7 is the bounded indicator $\mathbf 1_Z$, and the moments are uncentred.

Verification

abstract read (primary text, OUP page), 5 Sep 2026.