On Cauchy-Stieltjes Kernel Families

TL;DR

This paper investigates unique properties of Cauchy-Stieltjes families, focusing on their variance functions and conditions for extending their domain of means beyond the natural limits, highlighting differences from exponential families.

Contribution

It introduces new relationships between variance functions of iterated Cauchy-Stieltjes families and explores domain extension conditions.

Findings

Relation between variance function and pseudo-variance function

Conditions for extending the domain of means

Distinct properties from exponential families

Abstract

We explore properties of Cauchy-Stieltjes families that have no counterpart in exponential families. We relate the variance function of the iterated Cauchy-Stieltjes family to the pseudo-variance function of the initial Cauchy-Stieltjes family. We also investigate when the domain of means can be extended beyond the "natural domain".