## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 35, Number 3 (1964), 1059-1063.

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

T. W. Anderson and S. Das Gupta

#### 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 $\Sigma$.

#### Article information

**Source**

Ann. Math. Statist., Volume 35, Number 3 (1964), 1059-1063.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177703264

**Digital Object Identifier**

doi:10.1214/aoms/1177703264

**Mathematical Reviews number (MathSciNet)**

MR164407

**Zentralblatt MATH identifier**

0211.50403

**JSTOR**

links.jstor.org

#### Citation

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. 35 (1964), no. 3, 1059--1063. doi:10.1214/aoms/1177703264. https://projecteuclid.org/euclid.aoms/1177703264

#### Corrections

- See Correction: T. W. Anderson, S. Das Gupta. Correction Notes: Correction to "A Monotonicity Property of the Power Functions of Some Tests of the Equality of Two Covariance Matrices". Ann. Math. Statist., Volume 36, Number 4 (1965), 1318--1318.Project Euclid: euclid.aoms/1177700008