For two multivariate nonsingular normal distributions, the familiar null hypothesis of equal dispersion matrices is considered against various alternatives stated in terms of certain characteristic roots, and a physical interpretation is given for the alternatives considered. An inference procedure, which depends on similar regions and is based on one independent random sample from each of the two distributions, is proposed for the null hypothesis against each of the alternative hypotheses. Also, for three of the cases, conservative confidence bounds are obtained on one or more parametric functions which might be interpreted as measures of departure from the null hypothesis in the direction of the corresponding alternative.
"Two-Sample Comparisons of Dispersion Matrices for Alternatives of Intermediate Specificity." Ann. Math. Statist. 33 (2) 432 - 437, June, 1962. https://doi.org/10.1214/aoms/1177704569