Theoretical Statistics and Mathematics Unit, ISI Delhi

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]

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