发表机构
HIT – Holon Institute of Technology（霍隆理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对自然指数族的方差函数逆问题，证明当方差函数形如 \\(m^2G(m)\\) 且 \\(G\\) 为非负系数幂级数时，对应的自然指数族在正半轴上绝对连续，补充了计数情形的对偶结果，并给出累积量与统计推断推论。
AI 中文摘要
自然指数族（NEF）由其方差函数（VF）刻画，即由对子 \\((V,M)\\) 刻画，其中 \\(V\\) 将方差表示为均值的函数，\\(M\\) 是均值域。这一刻画自然引出了逆问题：给定函数 \\(V\\) 何时是某个 NEF 的 VF，以及族的哪些结构性质可以直接从 \\(V\\) 读出？Bar-Lev (1987) 作为关于绝对单调函数的更一般结果的一部分，证明了每个非零多项式 \\[ V(m)=\sum_{j=1}^{r} a_j m^j,\qquad a_j\ge 0, \\] 都是正均值域上无限可分 NEF 的 VF。该多项式类穷尽地分为两种情况：\\(a_1>0\\) 和 \\(a_1=0\\)。Bar-Lev, Letac 和 Ridder (2024) 证明了在缩放使得 \\(a_1=1\\) 后，第一种情况产生一个支撑在 \\(\mathbb N_0\\) 上的计数 NEF。我们建立了互补结果：当 \\(a_1=0\\) 时，相应的 NEF 关于正半轴上的 Lebesgue 测度绝对连续。更一般地，我们证明当 \\[ V(m)=m^2G(m), \\] 其中 \\(G\\) 是具有非负系数的非零幂级数时，绝对连续性成立。当 \\(G\\) 是多项式时，均值域是整个正半轴。我们还记录了关于累积量结构和统计推断的几个推论。
英文摘要
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.
