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

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 $P_t((-\infty ,t))\le \alpha \le P_t((-\infty ,t])$ 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 $Q_{-t}([0,t^{-1}))\le \alpha \le Q_{-t}([0,t^{-1}])$ 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 $t.$

arXiv comment7 pages