AISM 52, 1-14
(Received November 27, 1997; revised January 21, 1999)
Abstract. Consider the test problem about matrix normal mean M with the null hypothesis M=O against the alternative that M is nonnegative definite. In our previous paper (Kuriki (1993, Ann. Statist., 21, 1379-1384)), the null distribution of the likelihood ratio statistic has been given in the form of a finite mixture of chi2 distributions referred to as \bar{\chi}2 distribution. In this paper, we investigate differential-geometric structure such as second fundamental form and volume element of the boundary of the cone formed by real nonnegative definite matrices, and give a geometric derivation of this null distribution by virtue of the general theory on the \bar{\chi}2 distribution for piecewise smooth convex cone alternatives developed by Takemura and Kuriki (1997, Ann. Statist., 25, 2368-2387).
Key words and phrases: One-sided test for covariance matrices, symmetric cone, mixed volume, second fundamental form, volume element.