New characterizations of the Letac-Mora class of real cubic natural exponential families

TL;DR

This paper provides new characterizations of the Letac-Mora class of real cubic natural exponential families through properties involving Monge-Ampère equations, differential equations for variance functions, and prior distribution equalities.

Contribution

It introduces three equivalent properties that characterize the Letac-Mora class, extending previous properties known for Wishart and quadratic NEFs to cubic NEFs.

Findings

Cumulant functions satisfy Monge-Ampère equations

Variance functions obey specific differential equations

Equality of prior distribution families characterizes the class

Abstract

In this paper, we give three equivalent properties of the class of multivariate simple cubic natural exponential families (NEF's). The first property says that the cumulant function of any basis of the family is a solution of some Monge-Amp\'{e}re equation, the second is that the variance function satisfies a differential equation, and the third is characterized by the equality between two families of prior distributions related to the NEF. These properties represent the extensions to this class of the properties stated in $\cite{Casalis(1996)}$ and satisfied by the Wishart and the simple quadratic NEF's. We also show that in the real case, each of these properties provides a new characterization of the Letac-Mora class of real cubic NEF's.