General canonical correlations with applications to group symmetry models

In this paper, we define general canonical correlations, which generalize the canonical correlations developed by Hotelling, and general canonical covariate pairs, the corresponding linear statistic. We also define canonical variance distances with corresponding canonical distance variates. In a rather broad setting, these parameters and their corresponding linear statistics are characterized in terms of certain eigenvalues and eigenvectors. For seven of the ten group symmetry testing problems discussed in Andersson, Brّns, and Jensen (1983) [4], these are the eigenvalues used to represent the maximal invariant statistic, and additional observations regarding the canonical correlations are made for these testing problems.