Location and scale behaviour of the quantiles of a natural exponential family
share
Let P_0 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 P_0 is the Gaussian distribution N(μ_α,1). In other terms, if for all t, 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. t is always a median of P_t, has been considered in Letac et al. (2018). 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 distribution with parameters p and t.
READ FULL TEXT