A Monotonicity Property of the Power Functions of Some Tests of the Equality of Two Covariance Matrices
Anderson, T. W. ; Gupta, S. Das
Ann. Math. Statist., Tome 35 (1964) no. 4, p. 1059-1063 / Harvested from Project Euclid
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 $\Sigma$.
Publié le : 1964-09-14
Classification: 
@article{1177703264,
     author = {Anderson, T. W. and Gupta, S. Das},
     title = {A Monotonicity Property of the Power Functions of Some Tests of the Equality of Two Covariance Matrices},
     journal = {Ann. Math. Statist.},
     volume = {35},
     number = {4},
     year = {1964},
     pages = { 1059-1063},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1177703264}
}
Anderson, T. W.; Gupta, S. Das. A Monotonicity Property of the Power Functions of Some Tests of the Equality of Two Covariance Matrices. Ann. Math. Statist., Tome 35 (1964) no. 4, pp.  1059-1063. http://gdmltest.u-ga.fr/item/1177703264/