Estimation of the eigenvalues of [Sigma]1[Sigma]2-1

In the normal two-sample problem, an invariant test for the hypothesis of the equality of the population covariance matrices, H:[Sigma]1 = [Sigma]2 vs A:[Sigma]1 [not equal to] [Sigma]2, has a power function which depends only on the eigenvalues of [Sigma]1[Sigma]2-1. An orthogonally invariant minimax estimator of these eigenvalues is proposed which has very desirable properties. Namely, the estimated eigenvalues are always positive and they follow the same ordering as the eigenvalues of S1S2-1 calculated from the usual sample covariance matrices. Moreover, it has an explicit expression that can be easily calculated and yields substantial risk reductions.