The $2d+4$ simple quadratic natural exponential families on ${\bf R}\sp d$

Abstract

The present paper describes all the natural exponential families on $\mathbb{R}^d$ whose variance function is of the form $V(m) = am \otimes m + B(m) + C$, with $m \otimes m(\theta) = \langle \theta, m \rangle m$ and B linear in m. There are $2d + 4$ types of such families, which are built from particular mixtures of families of Normal, Poisson, gamma, hyperbolic on $\mathbb{R}^d$ and negative-multinomial distributions. The proof of this result relies mainly on techniques used in the elementary theory of Lie algebras.