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Dispersive ordering—Some applications and examples

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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.

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Authors and Affiliations

  1. Department of Statistics, Seoul National University, 151-742, Seoul, Korea

    Jongwoo Jeon

  2. Department of Mathematics and Statistics, Portland State University, 97201, Portland, Oregon, USA

    Subhash Kochar

  3. School of Mathematics and Statistics, Carleton University, K1S 5B6, Ottawa, Ontario, Canada

    Chul Gyu Park

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  1. Jongwoo Jeon
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  2. Subhash Kochar
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  3. Chul Gyu Park
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Additional information

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.

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Jeon, J., Kochar, S. & Park, C.G. Dispersive ordering—Some applications and examples. Statistical Papers 47, 227–247 (2006). https://doi.org/10.1007/s00362-005-0285-4

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