SOME GEOMETRY OF THE CONE OF NONNEGATIVE DEFINITE MATRICES AND WEIGHTS OF ASSOCIATED χ̄^{2} DISTRIBUTION

SATOSHI KURIKI^{1} AND AKIMICHI TAKEMURA^{2}

1The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan

2Faculty of Economics, University of Tokyo, 7-3-1 Bunkyo-ku, Tokyo 113-0033, Japan

(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 χ^{2} distributions referred to as χ̄^{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 χ̄^{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.