Characterizations of the beta distribution on symmetric matrices

Abstract

a b s t r a c tThis paper extends to the beta-Wishart distribution on symmetric matrices, two characterizations of the beta distributions on R, due to Seshadri and Wesolowski and based on some properties of constancy regression.

Characterization of beta distribution on symmetric cones

“…Our proof will heavily rely on the solution to the generalization of this equation to the cone Ω + , which was given in [9]. Similar characterization of beta distribution for random matrices was proved under numerous additional assumptions in [6]. The characterization of 2 × 2 matrix-variate beta distribution was also given by [2], but the characterization condition was of a different nature.…”

“…where Det denotes the determinant in the space of endomorphisms on Ω. Inserting a multiplication algorithm g = w(y), y ∈ Ω, and x = e we obtain Det (w(y)) = (det y) dim Ω/r (5) and hence det(w(y)x) = det y det x (6) for any x, y ∈ Ω.…”

Generalized Extended Matrix Variate Beta and Gamma Functions and Their Applications

“…For some recent advances the reader is refereed to Hassairi and Regaig [12], Farah and Hassairi [4], Gupta and Nagar [11], and Zine [25]. However, generalizations of the extended gamma and extended beta functions defined by (5) and (6), respectively, to the matrix case have not been defined and studied.…”

Kshirsagar–Tan independence property of beta matrices and related characterizations

“…Our general goal is to develop characterizations related to independence properties of matrix variate beta distributions. A characterization of this type, related to matrix versions of neutrality properties, has recently been given by Hassairi and Regaig (2006). They extended to matrix random variables the result for univariate beta laws obtained in Seshadri and Weso lowski (2003).…”