Dispersive ordering - Some applications and examples

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 <=(disp) X) if G(-1)(beta) - G(-1)(alpha) <= F-1(beta) - F-1(alpha), for all 0 < alpha <= beta < 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 <=(disp) X is that vertical bar Y-1 - Y(2)vertical bar is stochastically smaller than vertical bar X-1 - X(2)vertical bar and this in turn implies var(Y) : var(X) as well as E[vertical bar Y-1 - Y(2)vertical bar] <= E[X-1 - X(2)vertical bar], where X-1, X-2 (Y-1, Y-2) 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.