# Publications and Preprints

Dispersive Ordering - Some Applications and Examples
by
Jongwoo Jeon, Subhash Kochar and Chul Gyu Park
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 {\it dispersed} than $X$ ( $Y\le_{disp} X$) if $G^{-1} (\beta) - G^{-1} (\alpha)\le F^{-1} (\beta) - F^{-1} (\alpha)$, for all $0 < \alpha \le \beta < 1$. 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 $Y\le_{disp} X$ is that $|Y_1 -Y_2|$ is stochastically smaller than $|X_1 - X_2|$ and this in turn implies $var(Y) \le var(X)$ as well as $E[|Y_1 - Y_2|] \le E[|X_1 -X_2|]$, 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.

isid/ms/2004/03 [fulltext]