Volume 24, 2020
|Page(s)||244 - 251|
|Published online||24 March 2020|
Location and scale behaviour of the quantiles of a natural exponential family
Dipartimento di Matematica, Sapienza Università di Roma,
2 Faculty of Mathematics and Information Sciences, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland.
3 Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 16 May 2019
Let P0 be a probability on the real line generating a natural exponential family (Pt)t∈ℝ. Fix α in (0, 1). We show that the property that Pt((−∞, t)) ≤ α ≤ Pt((−∞, t]) for all t implies that there exists a number μα such that P0 is the Gaussian distribution N(μα, 1). In other terms, if for all t, the number t is a quantile of Pt 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 Pt, 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−t)t>0. We show that Q−t([0, t−1)) ≤ α ≤ Q−t([0, t−1]) for all t > 0 implies that there exists a number p = pα > 0 such that Q(dx) ∝ xp−1dx, and thus Q−t has to be a gamma law with parameters p and t.
Mathematics Subject Classification: 62E10 / 60E05 / 45E10
Key words: Characterization of normal and gamma laws / one-dimensional exponential families / quantiles of a distribution / Deny equations
© The authors. Published by EDP Sciences, SMAI 2020
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