TL;DR: In this article, the classical family of Wishart distributions on a cone of positive definite matrices and its fundamental features are extended to a family of generalized Wishart distribution on a homogeneous cone using the theory of exponential families.
Abstract: The classical family of Wishart distributions on a cone of positive definite matrices and its fundamental features are extended to a family of generalized Wishart distributions on a homogeneous cone using the theory of exponential families. The generalized Wishart distributions include all known families of Wishart distributions as special cases. The relationships to graphical models and Bayesian statistics are indicated.