A Monotonicity Property of the Power Functions of Some Invariant Tests for MANOVA

Abstract

The main result of the current research describes a monotonicity property of certain invariant tests for the multivariate analysis of variance problem. Suppose $X: r \times p$ has a normal distribution, $EX = \Theta$ and the rows of $X$ are independent, each with unknown covariance matrix $\Sigma: p \times p$. Let $S = p \times p$ have a Wishart distribution $W(\Sigma, p, n), S$ independent of $X$. If $K$ is the acceptance region of an invariant test for the null hypothesis $\Theta = 0$, let $\rho_K(\delta)$ denote the power function of $K$, where $\delta = (\delta_1, \cdots, \delta_t), t \equiv \min (r, p)$ and $\delta_1^2,\cdots,\delta_t^2$ are the $t$ largest characteristic roots of $\Theta\sigma^{-1}\Theta'$. A main result is THEOREM. If $K$ is a convex set (in $(X, S))$, then $\rho_K(\delta)$ is a Schur-convex function of $\delta$. Standard tests to which the above theorem can be applied include the Roy maximum root test and the Lawley-Hotelling trace test.