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

Abstract

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.