Stein's general technique for improving upon the best invariant unbiased and minimax estimators of the normal covariance matrix is described. The technique is to obtain solutions to a certain differential inequality involving the eigenvalues of the sample covariance matrix. Several improved estimators are obtained by solving the differential inequality. These estimators shrink or expand the sample eigenvalues depending on their magnitude. A scale invariant, adaptive minimax estimator is also obtained.
"Estimation of a Covariance Matrix under Stein's Loss." Ann. Statist. 13 (4) 1581 - 1591, December, 1985. https://doi.org/10.1214/aos/1176349756