Riesz exponential families on homogeneous cones

TL;DR

This paper generalizes exponential families on homogeneous cones by defining new power functions, extending Gindikin's result, and characterizing when these generate valid statistical models with explicit variance functions.

Contribution

It introduces generalized power functions on homogeneous cones, extends Gindikin's theorem to these functions, and characterizes the resulting exponential families and their variance functions.

Findings

Identified conditions for multipliers to produce Laplace transforms of positive measures.

Characterized when these measures generate exponential families.

Calculated the variance functions of the new exponential families.

Abstract

In this paper, we introduce, for a multiplier $\chi$, a notion of generalized power function $x\mapsto \Delta_{\chi}(x),$ defined on the homogeneous cone ${\mathcal{P}}$ of a Vinberg algebra ${\mathcal{A}}$. We then extend to ${\mathcal{A}}$ the famous Gindikin result, that is we determine the set of multipliers $\chi$ such that the map $\theta \mapsto \Delta_{\chi}(\theta ^{-1})$, defined on ${\mathcal{P}}^{\ast}$, is the Laplace transform of a positive measure $R_{\chi}$. We also determine the set of $\chi $ such that $R_{\chi}$ generates an exponential family, and we calculate the variance function of this family