Abstract
The Koszul–Vinberg characteristic function plays a fundamental role in the theory of convex cones. We give an explicit description of the function and related integral formulas for a new class of convex cones, including homogeneous cones and cones associated with chordal (decomposable) graphs appearing in statistics. Furthermore, we discuss an application to maximum likelihood estimation for a certain exponential family over a cone of this class.
Highlights
Let Ω be an open convex cone in a vector space Z
We present a wide class of cones, including both of them, and give an explicit expression of the Koszul–Vinberg characteristic function (Section 3)
We present a generalization of matrix realization dealing with a wide class of convex cones, which turns out to include cones associated with chordal graphs
Summary
Let Ω be an open convex cone in a vector space Z. We present a wide class of cones, including both of them, and give an explicit expression of the Koszul–Vinberg characteristic function (Section 3). Based on such results, our matrix realization method [15,17,18] has been developed for the purpose of the efficient study of homogeneous cones. We present a generalization of matrix realization dealing with a wide class of convex cones, which turns out to include cones associated with chordal graphs. It was an enigma for the author that some formulas in [11,19]. The identity matrix of size p is denoted by I p
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