Wishart Distributions on Homogeneous Cones

Explicit Formula of Koszul–Vinberg Characteristic Functions for a Wide Class of Regular Convex Cones

“…which means (24). When q 1 > 0, the Euclidean measure dx equals 2 q 1 /2 x q 1 11 dx 11 dũdx by the change of variables in (6).…”

“…The notion of generalized power functions, as well as the Γ-type integral formulas are due to Gindikin [8] (see also [23]). The Wishart distributions for homogeneous cones are studied in [17,21,24,25].…”

Realization of Homogeneous Cones Through Oriented Graphs

“…Moreover, we have used Lemma 2.2 and (2.4) to obtain t (ρ j (ξ ))y = y ξ (y ∈ V rj ). Note that ϕ [1] (x) for x ∈ V is nothing but ϕ(x) expressed in a different way of operator matrix from (3.3). Here we need the following two lemmas.…”

“…However, in practice, it requires some expertise to treat such nonassociative matrix algebras. In view of recent applications [1], [4] and [12] of homogeneous cones to statistics, it is desirable to have a description of homogeneous cones with easier access. This is the first motivation of our study.…”

“…Note that multiplying the matrix in (1.4) by the 4 × 4 matrix diag (1), 0 1 1 0 , (1) from the left and the right, we see that the description (1.4) is transformed to (4.2). Needless to say, the presentation (1.4) looks like that of (1.2), and is better than that of (1.3).…”