Wishart laws and variance function on homogeneous cones

TL;DR

This paper systematically studies Wishart laws on homogeneous cones, explicitly computing the inverse mean map and variance function of their exponential families, advancing understanding of their mathematical properties.

Contribution

It provides explicit formulas for the inverse mean map and variance function of Wishart exponential families on homogeneous cones, a novel contribution in this area.

Findings

Explicit inverse mean map derived for Wishart laws.

Variance function explicitly computed for Wishart exponential families.

Enhanced understanding of the structure of Wishart laws on homogeneous cones.

Abstract

We present a systematic study of Riesz measures and their natural exponential families of Wishart laws on a homogeneous cone. We compute explicitly the inverse of the mean map and the variance function of a Wishart exponential family.