AISM 52, 1-14

## Some geometry of the cone of nonnegative definite matrices and weights of associated \bar{\chi}^2 distribution

### Satoshi Kuriki^{1} and Akimichi Takemura^{2}

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

^{2}Faculty 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 *chi*^{2} 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.

**Source**
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