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)=∑j=1r aj mj, aj≥ 0, is the VF of an infinitely divisible NEF on a positive mean domain. This polynomial class splits exhaustively into the cases a1>0 and a1=0. Bar-Lev, Letac and Ridder (2024) proved that, after scaling so that a1=1, the first case yields a counting NEF supported on ℕ0. We establish the complementary result: when a1=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)=m2G(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.
Related
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BibTeX
@misc{barlev2026from,
title = {{From Counting to Continuous Natural Exponential Families}},
author = {Shaul K. Bar-Lev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05478},
eprint = {2610.05478},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05478},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Counting to Continuous Natural Exponential Families
AU - Shaul K. Bar-Lev
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05478
UR - https://arxiv.org/abs/2610.05478
AB - 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)=∑_j=1^r a_j m^j, a_j≥ 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 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.
ER -