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

¹ Dipartimento di Matematica, Sapienza Università di Roma, 00185 Roma, Italy.
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: gerard.letac@math.univ-toulouse.fr

Received: 4 February 2019
Accepted: 16 May 2019

Abstract

Let P₀ be a probability on the real line generating a natural exponential family (P_t)_t∈ℝ. Fix α in (0, 1). We show that the property that P_t((−∞, t)) ≤ α ≤ P_t((−∞, t]) for all t implies that there exists a number μ_α such that P₀ is the Gaussian distribution N(μ_α, 1). In other terms, if for all t, the number t is a quantile of P_t 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_t, 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 = p_α > 0 such that Q(dx) ∝ x^p−1dx, and thus Q_−t has to be a gamma law with parameters p and t.