Stochastic Dominance and Moments of Distributions

Stochastic dominance orders of all finite degrees are defined on the set of distribution functions on the nonnegative real numbers in terms of integrals of the distributions. It is proved that if F strictly nth-degree stochastically dominates G, and if the moments of F and G through order n are finite with μ F k = ∫X k dF (x), then (μ F 1 , …, μ F n ) ≠ (μ G 1 , …, μ G n ) and (−1) k−1 μ F k > (−1) k−1 μ G k for the smallest k for which μ F k ≠ μ G k .