From Counting to Continuous Natural Exponential Families

Abstract

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 \[ V(m)=\sum_{j=1}^{r} a_j m^j,\qquad a_j\ge 0, \] 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 \[ V(m)=m^2G(m), \] 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.