Bivariate natural exponential families with quadratic diagonal of the variance function

TL;DR

This paper characterizes bivariate natural exponential families with a specific quadratic form for the diagonal of their variance function, identifying conditions for these families to correspond to Laplace transforms of probability measures.

Contribution

It provides a complete characterization of bivariate NEFs with quadratic diagonal variance functions, expanding understanding of their structure and conditions.

Findings

Derived explicit conditions for the variance function form

Identified when these families correspond to probability measures

Extended the classification of natural exponential families

Abstract

We characterize bivariate natural exponential families having the diagonal of the variance function of the form \[ \textrm{diag} V(m_1,m_2)=\left(Am_1^2+am_1+bm_2+e,Am_2^2+cm_1+dm_2+f\right), \] with $A<0$ and $a,\ldots,f\in\mathbb{R}$. The solution of the problem relies on finding the conditions under which a specific parametric family of functions consists of Laplace transforms of some probability measures.