Cumulative distribution functions for the five simplest natural exponential families

TL;DR

This paper characterizes the five simplest natural exponential families—binomial, Poisson, negative binomial, Gaussian, and Gamma—using properties of their cumulative distribution functions and Moebius transformations.

Contribution

It provides a novel characterization of these five exponential families based on integral representations of their CDFs and invariance under Moebius functions.

Findings

Characterization of discrete families: binomial, Poisson, negative binomial.

Characterization of continuous families: Gaussian, Gamma.

Use of Moebius functions and cross ratio invariance in proofs.

Abstract

Suppose that the distribution of $X_a$ belongs to a natural exponential family concentrated on the nonegative integers and is such that $\E(z^{X_a})=f(az)/f(a)$. Assume that $\Pr(X_a\leq k)$ has the form $c_k\int_a ^{\infty}u^k\mu(du)$ for some number $c_k$ and some positive measure $\mu,$ both independent of $a.$ We show that this asumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family. Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamma. Ultimately, the proofs rely on the fact that only Moebius functions preserve the cross ratio, \textsc{Keywords:} Binomial, Poisson and negative binomial distributions. Gaussian and Gamma distributions. Moebius transforms. Cross ratio.