A Monotonicity Property of the Power Functions of Some Tests of the Equality of Two Covariance Matrices

Abstract

Invariant tests of the hypothesis that $\mathbf\Sigma_1 = \Sigma_2$ are based on the characteristic roots of $S_1S^{-1}_2$, say $c_1 \geqq c_2 \geqq \cdots \geqq c_p$, where $\Sigma_1$ and $\Sigma_2$ and $\mathbf{S}_1$ and $\mathbf{S}_2$ are the population and sample covariance matrices, respectively, of two multivariate normal populations; the power of such a test depends on the characteristic roots of $\Sigma_1\Sigma^{-1}_2$. It is shown that the power function is an increasing function of each ordered root of $\Sigma_1\Sigma^{-1}_2$ if the acceptance region of the test has the property that if $(c_1, \cdots, c_p)$ is in the region then any point with coordinates not greater than these, respectively, is also in the region. Examples of such acceptance regions are given. For testing the hypothesis that $\Sigma = I$, a similar sufficient condition is derived for a test depending on the roots of a sample covariance matrix $\mathbf{S}$, based on observations from a normal distribution with covariance matrix $\Sigma$, to have the power function monotonically increasing in each root of $\Sigma$.