Natural Exponential Families: Resolution of A Conjecture and Existence of Reduction Functions

Abstract

One-parameter natural exponential family (NEF) plays fundamental roles in probability and statistics. This article contains two independent results: (a) A conjecture of Bar-Lev, Bshouty and Enis states that a polynomial with a simple root at $0$ and a complex root with positive imaginary part is the variance function of some NEF with mean domain $\left(0,\infty\right)$ if and only if the real part of the complex root is not positive. This conjecture is resolved. The positive answer to this conjecture enlarges existing family of polynomials that are able to generate NEFs, and it helps prevent practitioners from choosing incompatible functions as variance functions for statistical modeling using NEFs. (b) if a random variable $ξ$ has parametric distributions that form a infinitely divisible NEF whose induced measure is absolutely continuous with respect to its basis measure, then there exists a deterministic function $h$, called "reduction function", such that $\mathbb{E} \left(h\left(ξ\right)\right)=\mathbb{V}\left(ξ\right)$, i.e., $h\left(ξ\right)$ is an unbiased estimator of the variance of $ξ$. The reduction function has applications to estimating latent, low-dimensional structures and to dimension reduction in the first and/or second moments in high-dimensional data.