An Exact Decomposition Theorem and a Unified View of Some Related Distributions for a Class of Exponential Transformation Models on Symmetric Cones

Abstract

A class of exponential transformation models is defined on symmetric cones $\Omega$ with the group of automorphisms on $\Omega$ as the acting group. We show that these models are reproductive and the exponent of their joint distribution for a given sample of size $q$ can be split into $q$ independent components, introducing one sample point at a time. The automorphism group can be factorized into the group of positive dilation and another group. Accordingly, the symmetric cone $\Omega$ can be seen as the direct product of $\mathbb{R}^+$ and a unit orbit, and every $x$ in $\Omega$ can be identified by its orbital decomposition. We derive the distributions of the independent components of the exponent, of the "length" of $x$, the "direction" of $x$, the conditional distribution of the direction given the length and other distributions for a given sample. The Wishart distribution and the hyperboloid distribution are two special cases in the class we define. We also give a unified view of several distributions which are usually treated separately.