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September, 1964 A Monotonicity Property of the Power Functions of Some Tests of the Equality of Two Covariance Matrices
T. W. Anderson, S. Das Gupta
Ann. Math. Statist. 35(3): 1059-1063 (September, 1964). DOI: 10.1214/aoms/1177703264

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 $\Sigma$.

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T. W. Anderson. S. Das Gupta. "A Monotonicity Property of the Power Functions of Some Tests of the Equality of Two Covariance Matrices." Ann. Math. Statist. 35 (3) 1059 - 1063, September, 1964. https://doi.org/10.1214/aoms/1177703264

Information

Published: September, 1964
First available in Project Euclid: 27 April 2007

zbMATH: 0211.50403
MathSciNet: MR164407
Digital Object Identifier: 10.1214/aoms/1177703264

Rights: Copyright © 1964 Institute of Mathematical Statistics

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Vol.35 • No. 3 • September, 1964
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