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September, 1974 A Monotonicity Property of the Power Functions of Some Invariant Tests for MANOVA
Morris L. Eaton, Michael D. Perlman
Ann. Statist. 2(5): 1022-1028 (September, 1974). DOI: 10.1214/aos/1176342821


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.


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Morris L. Eaton. Michael D. Perlman. "A Monotonicity Property of the Power Functions of Some Invariant Tests for MANOVA." Ann. Statist. 2 (5) 1022 - 1028, September, 1974.


Published: September, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0287.62028
MathSciNet: MR391394
Digital Object Identifier: 10.1214/aos/1176342821

Primary: 62H15

Keywords: $G$-orbit , convex set , invariant tests , MANOVA , Monotonicity , power function , Schur-convex

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 5 • September, 1974
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