Nonnegativity of odd functional moments of positive random variables with decreasing density

Abstract

In this note we give some results on the nonnegativity of odd functional moments of random variables with a decreasing density. More precisely, we prove by purely elementary arguments, Egf( X − EX) ⩾ 0 for suitable functions gf that satisfy gf( x) = − gf(− x) for all x ⩾ 0 and random variables X ⩾ 0 with a decreasing Lebesgue density on (0, ∞) or counting density on N 0. The motivation came from a problem recently published in Statistica Neerlandica (Vol. 43, p. 66). We give a more specialized result in this paper.