From Counting to Continuous Natural Exponential Families

Natural exponential families (NEFs) are characterized by their variance functions (VFs), namely by the pair $(V,M)$, where $V$ expresses the variance as a function of the mean and $M$ is the mean domain. This characterization naturally raises the inverse question: when is a given function $V$ the VF of an NEF, and what structural properties of the family can be read directly from $V$? Bar-Lev (1987), as part of a more general result for absolutely monotone functions, showed that every nonzero polynomial $$V(m)=\sum _{j=1}^r{a}_j{m}^j,a_j\ge 0,$$ is the VF of an infinitely divisible NEF on a positive mean domain. This polynomial class splits exhaustively into the cases $a_1>0$ and $a_1=0$. Bar-Lev, Letac and Ridder (2024) proved that, after scaling so that $a_1=1$, the first case yields a counting NEF supported on ${\mathbb{N}}_0$. We establish the complementary result: when $a_1=0$, the corresponding NEF is absolutely continuous with respect to Lebesgue measure on the positive half-line. More generally, we prove absolute continuity whenever $$V(m)=m^{2}G(m),$$ where $G$ is a nonzero power series with nonnegative coefficients. When $G$ is a polynomial, the mean domain is the entire positive half-line. We also record several consequences for the cumulant structure and for statistical inference.