## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 33, Number 4 (1962), 1463-1465.

### On a Property of a Test for the Equality of Two Normal Dispersion Matrices Against One-sided Alternatives

#### Abstract

The purpose of this paper is to establish the monotonicity property of some tests suggested by Roy and Gnanadesikan [2] for the problem of testing the null hypothesis of equality of two dispersion matrices against some specific alternatives. If $\Sigma_1$ and $\Sigma_2$ denote the dispersion matrices of two non-singular $p$-variate normals and $\gamma_1, \gamma_2, \cdots, \gamma_p$ denote the characteristic roots (all positive) of $\Sigma_1\Sigma^{-1}_2$, then the null hypothesis is $H_0$: all $\gamma_i$'s are equal to unity. The alternative hypotheses to be considered are: (i) $H_1 : \gamma_m > 1$; (ii) $H_2 : \gamma_M < 1$; (iii) $H_3 : \gamma_M > 1$; (iv) $H_4 : \gamma_m < 1$, where $\gamma_m$ and $\gamma_M$ denote, respectively, the smallest and the largest of the $\gamma_i$. Let us denote the largest and smallest characteristic roots of any square matrix $A$ by $\operatorname{ch}_{\max} (A)$ and $\operatorname{ch}_{\min} (A)$, respectively.

#### Article information

**Source**

Ann. Math. Statist., Volume 33, Number 4 (1962), 1463-1465.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177704380

**Digital Object Identifier**

doi:10.1214/aoms/1177704380

**Mathematical Reviews number (MathSciNet)**

MR141191

**Zentralblatt MATH identifier**

0136.40105

**JSTOR**

links.jstor.org

#### Citation

Mikhail, Wadie F. On a Property of a Test for the Equality of Two Normal Dispersion Matrices Against One-sided Alternatives. Ann. Math. Statist. 33 (1962), no. 4, 1463--1465. doi:10.1214/aoms/1177704380. https://projecteuclid.org/euclid.aoms/1177704380