Location and scale behaviour of the quantiles of a natural exponential family

TL;DR

This paper characterizes when the quantiles of natural exponential families imply Gaussian or gamma distributions, showing that certain quantile behaviors uniquely determine these distributions.

Contribution

It proves that specific quantile conditions across the family imply the underlying distribution must be Gaussian or gamma, providing a characterization of these distributions.

Findings

Quantile conditions characterize Gaussian distributions in exponential families.

Quantile conditions characterize gamma distributions in related exponential families.

The results extend previous median-based characterizations to general quantiles.

Abstract

Let $P_0$ be a probability on the real line generating a natural exponential family $(P_t)_{t\in \mathbb {R}}$. Fix $\alpha$ in $ (0,1).$ We show that the property that $P_t((-\infty,t)) \leq \alpha \leq P_t((-\infty,t])$ for all $t$ implies that there exists a number $\mu_\alpha$ such that $P_0$ is the Gaussian distribution $N(\mu_{\alpha},1).$ In other terms, if for all $t$, $t$ is a quantile of $P_t$ associated to some threshold $\alpha\in (0,1)$, then the exponential family must be Gaussian. The case $\alpha=1/2$, \textit{i.e.} $t$ is always a median of $P_t,$ has been considered in Letac \textit{et al.} (2018). Analogously let $Q$ be a measure on $[0,\infty)$ generating a natural exponential family $(Q_{-t})_{t>0}$. We show that $Q_{-t}([0,t^{-1}))\leq \alpha \leq Q_{-t}([0,t^{-1}])$ for all $t>0$ implies that there exists a number $p=p_{\alpha}>0$ such that $Q(dx)\propto x^{p-1}dx,$ and thus $Q_{-t}$ has to be a gamma distribution with parameters $p$ and $t.$