We consider the problem of estimating parameter matrices which occur in the noncentral Wishart, noncentral multivariate $F$ and canonical correlations distributions. A decision-theoretic approach is taken with squared error as the loss function. In these three settings the eigenvalues of the parameter matrices are of primary interest. Sensible estimates of these are obtained by restricting attention to orthogonally invariant estimates of the parameter matrices, whose eigenvalues are functions only of sample eigenvalues.
"Estimation of Parameter Matrices and Eigenvalues in MANOVA and Canonical Correlation Analysis." Ann. Statist. 15 (4) 1651 - 1666, December, 1987. https://doi.org/10.1214/aos/1176350616