On monotonicity of the modified likelihood ratio test for the equality of two covariances

For testing the hypothesis of equality of two covariances (Σ₁ and Σ₂) of two p-dimensional multivariate normal populations, it is shown that the power function of the modified likelihood ratio test increases as λ₁ increases from one and λ_r decreases from one where λ₁ > … > λ_r > 0 are the distinct characteristic roots of Σ₁Σ₂^?1, r ≤ p. As a by-product we get the unbiased result already established by Sugiura and Nagao (1968).