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

### Mauro Piccioni , Bartosz Kołodziejek , Gerard Letac

#### Abstract

Let $P_0$ be a probability on the real line generating a natural exponential family $(P_t)_{t in R}$. Fix $alpha$ in $(0,1)$. We show that the property that $P_t((-infty,t)) <= alpha <= P_t((-infty,t])$ 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 $t$, the number $t$ is a quantile of $P_t$ associated to some threshold $alpha in (0,1)$, then the exponential family must be Gaussian. The case $alpha=1/2$, i.e. when $t$ is always a median of $P_t,$ has been considered in Letac et al. (2018). Analogously let $Q$ be a measure on $[0,infty)$ generating a natural exponential family $(Q_{-t})_{t>0}$. We show that $Q_{-t}([0,t^{-1}))<=alpha <= Q_{-t}([0,t^{-1}])$ 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 law with parameters $p$ and $t.$
 Author Mauro Piccioni Mauro Piccioni,, - , Bartosz Kołodziejek (FMIS / DPMS) Bartosz Kołodziejek,, - Department of Probability and Mathematical Statistics , Gerard Letac Gerard Letac,, - Journal series ESAIM-Probability and Statistics, [ESAIM - Probability and Statistics], ISSN 1292-8100, e-ISSN 1262-3318 Issue year 2020 Vol 24 Pages 244-251 Publication size in sheets 0.5 Keywords in Polish Charakteryzacje rozkładów normalnego i gamma, jednowymiarowe rodziny wykładniczy, kwantyle, równania Deny′ego Keywords in English Characterization of normal and gamma laws, one-dimensional exponential families, quantiles of a distribution, Deny equations. ASJC Classification 2613 Statistics and Probability Abstract in Polish W pracy przedstawiono charakteryzację rozkładów normalnego i gamma poprzez własności funkcji kwantylowych w jednowymiarowych rodzinach wykładniczych. Dowody głównych twierdzeń opierają się na równaniach Choqet′a-Deny′ego. DOI DOI:10.1051/ps/2019009 URL https://www.esaim-ps.org/articles/ps/abs/2020/01/ps190021/ps190021.html Language en angielski Score (nominal) 70 Score source journalList Score Ministerial score = 70.0, 23-09-2020, ArticleFromJournal Publication indicators Scopus Citations = 0; Scopus SNIP (Source Normalised Impact per Paper): 2018 = 0.783; WoS Impact Factor: 2018 = 0.652 (2) - 2018=0.763 (5) Citation count*
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