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 χ2 distributions referred to as X2 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 X2 distribution for piecewise smooth convex cone alternatives developed by Takemura and Kuriki (1997, Ann. Statist., 25, 2368–2387).
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Kuriki, S., Takemura, A. Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X2 Distribution. Annals of the Institute of Statistical Mathematics 52, 1–14 (2000). https://doi.org/10.1023/A:1004191713375
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DOI: https://doi.org/10.1023/A:1004191713375