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

For testing the hypothesis of equality of two covariances ([Sigma]1 and [Sigma]2) of two p-dimensional multivariate normal populations, it is shown that the power function of the modified likelihood ratio test increases as [lambda]1 increases from one and [lambda]r decreases from one where [lambda]1 > ... > [lambda]r > 0 are the distinct characteristic roots of [Sigma]1[Sigma]2-1, r <= p. As a by-product we get the unbiased result already established by [3].