Stochastic ordering of order statistics

Barbour, A D; Lindvall, T; Rogers, L (1991). Stochastic ordering of order statistics. Journal of Applied Probability, 28(2):278-286.

Abstract

If X<sub>i</sub>, i = 1, ⋯, n are independent exponential random variables with parameters λ<sub>1</sub>, ⋯, λ<sub>n</sub>, and if Y<sub>i</sub>, i = 1, ⋯, n are independent exponential random variables with common parameter equal to (λ<sub>1</sub> + ⋯ + λ<sub>n</sub>)/n then there is a monotone coupling of the order statistics X<sub>(1)</sub>, ⋯, X<sub>(n)</sub> and Y<sub>(1)</sub>, ⋯, Y<sub>(n)</sub>; that is, it is possible to construct on a common probability space random variables X<sub>i</sub><sup>′</sup>, Y<sub>i</sub><sup>′</sup>, i = 1, ⋯, n such that for each i, Y<sub>(i)</sub><sup>′</sup> ≤ X<sub>(i)</sub><sup>′</sup> a.s., where the law of the X<sub>i</sub><sup>′</sup> (respectively, the Y<sub>i</sub><sup>′</sup>) is the same as the law of the X<sub>i</sub> (respectively, the Y<sub>i</sub>) This result is due to Proschan and Sethuraman, and independently to Ball. We shall here prove an extension to a more general class of distributions for which the failure rate function r(x) is decreasing, and xr(x) is increasing. This very strong order relation allows comparison of properties of epidemic processes where rates of infection are not uniform with the corresponding properties for the homogeneous case. We further prove that for a sequence Z<sub>i</sub>, i = 1, ⋯, n of independent random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than the order statistics of a sample of n independent exponential random variables of mean n, but that the strong monotone coupling referred to above is impossible in general.

Abstract

If X<sub>i</sub>, i = 1, ⋯, n are independent exponential random variables with parameters λ<sub>1</sub>, ⋯, λ<sub>n</sub>, and if Y<sub>i</sub>, i = 1, ⋯, n are independent exponential random variables with common parameter equal to (λ<sub>1</sub> + ⋯ + λ<sub>n</sub>)/n then there is a monotone coupling of the order statistics X<sub>(1)</sub>, ⋯, X<sub>(n)</sub> and Y<sub>(1)</sub>, ⋯, Y<sub>(n)</sub>; that is, it is possible to construct on a common probability space random variables X<sub>i</sub><sup>′</sup>, Y<sub>i</sub><sup>′</sup>, i = 1, ⋯, n such that for each i, Y<sub>(i)</sub><sup>′</sup> ≤ X<sub>(i)</sub><sup>′</sup> a.s., where the law of the X<sub>i</sub><sup>′</sup> (respectively, the Y<sub>i</sub><sup>′</sup>) is the same as the law of the X<sub>i</sub> (respectively, the Y<sub>i</sub>) This result is due to Proschan and Sethuraman, and independently to Ball. We shall here prove an extension to a more general class of distributions for which the failure rate function r(x) is decreasing, and xr(x) is increasing. This very strong order relation allows comparison of properties of epidemic processes where rates of infection are not uniform with the corresponding properties for the homogeneous case. We further prove that for a sequence Z<sub>i</sub>, i = 1, ⋯, n of independent random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than the order statistics of a sample of n independent exponential random variables of mean n, but that the strong monotone coupling referred to above is impossible in general.

Statistics

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