Dispersive ordering—Some applications and examples

Abstract

A basic concept for comparing spread among probability distributions is that of dispersive ordering. Let X and Y be two random variables with distribution functions F and G, respectively. Let F−1 and G−1 be their right continuous inverses (quantile functions). We say that Y is less dispersed than X (Y≤dispX) if G−1(β)−G−1(α)≤F−1(β)−F−1(α), for all 0<α≤β<1. This means that the difference between any two quantiles of G is smaller than the difference between the corresponding quantiles of F. A consequence of Y≤dispX is that |Y1−Y2| is stochastically smaller than |X1−X2| and this in turn implies var(Y)≤var(X) as well as E[|Y1−Y2|]≤E[|X1−X2|], where X1, X2 (Y1, Y2) are two independent copies of X(Y). In this review paper, we give several examples and applications of dispersive ordering in statistics. Examples include those related to order statistics, spacings, convolution of non-identically distributed random variables and epoch times of non-homogeneous Poisson processes.