Dispersive ordering—Some applications and examples |
| |
Authors: | Jongwoo Jeon Subhash Kochar Chul Gyu Park |
| |
Institution: | (1) Department of Statistics, Seoul National University, 151-742 Seoul, Korea;(2) Department of Mathematics and Statistics, Portland State University, 97201 Portland, Oregon, USA;(3) School of Mathematics and Statistics, Carleton University, K1S 5B6 Ottawa, Ontario, Canada |
| |
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≤
disp
X) 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≤
disp
X is that |Y
1−Y
2| is stochastically smaller than |X
1−X
2| and this in turn implies var(Y)≤var(X) as well as E|Y
1−Y
2|]≤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.
This work was supported in part by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.
Subhash Kochar is thankful to Dr. B. Khaledi for many helpful discussions. |
| |
Keywords: | Exponential distribution proportional hazard rates hazard rate ordering Schur functions majorization and p-larger ordering convolutions parallel systems gamma distribution t-distribution |
本文献已被 SpringerLink 等数据库收录! |
|