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.