Location and scale behaviour of the quantiles of a natural exponential family
ESAIM: Probability and Statistics, Tome 24 (2020), pp. 244-251.

Let P0 be a probability on the real line generating a natural exponential family (P$$)$$. Fix α in (0, 1). We show that the property that P$$((−∞, t)) ≤ α ≤ P$$((−∞, t]) for all t implies that there exists a number μ$$ such that P0 is the Gaussian distribution N(μ$$, 1). In other terms, if for all t, the number t is a quantile of P$$ associated to some threshold α ∈ (0, 1), then the exponential family must be Gaussian. The case α = 1∕2, i.e. when t is always a median of P$$, has been considered in Letac et al. [Statist. Prob. Lett. 133 (2018) 38–41]. Analogously let Q be a measure on [0, ∞) generating a natural exponential family (Q$$)$$. We show that Q$$([0, t−1)) ≤ α ≤ Q$$([0, t−1]) for all t > 0 implies that there exists a number p = p$$ > 0 such that Q(dx) ∝ x$$dx, and thus Q$$ has to be a gamma law with parameters p and t.

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DOI : 10.1051/ps/2019009
Classification : 62E10, 60E05, 45E10
Mots-clés : Characterization of normal and gamma laws, one-dimensional exponential families, quantiles of a distribution, Deny equations 
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     author = {Piccioni, Mauro and Ko{\l}odziejek, Bartosz and Letac, G\'erard},
     title = {Location and scale behaviour of the quantiles of a natural exponential family},
     journal = {ESAIM: Probability and Statistics},
     pages = {244--251},
     publisher = {EDP-Sciences},
     volume = {24},
     year = {2020},
     doi = {10.1051/ps/2019009},
     mrnumber = {4079212},
     zbl = {1440.62053},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2019009/}
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Piccioni, Mauro; Kołodziejek, Bartosz; Letac, Gérard. Location and scale behaviour of the quantiles of a natural exponential family. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 244-251. doi : 10.1051/ps/2019009. http://archive.numdam.org/articles/10.1051/ps/2019009/

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