Existence of a normal scale mixture with a given variance and a percentile

Abstract

We consider the constrained normal scale mixture distribution family ℱ( p, Q; σ 2) = F is a normal scale mixture and F(Q) = p, Var F ( X) = σ 2. We show that ℱ( p,Q; σ 2) is nonempty, i.e., a normal scale mixture distribution with 100 p-th percentile at Q and variance = σ 2 exists if and only if p ⩾ a fixed constant. We next match the percentile Q with a normal percentile, Q N = σΦ −1( p). We prove that the class ℱ( p, Q N; σ 2) contains distributions other than N(0, σ 2) if and only if Q N/σ > 1.1906