Stochastic ordering of order statistics

Abstract

If X_i, i = 1, ⋯, n are independent exponential random variables with parameters λ₁, ⋯, λ_n, and if Y_i, i = 1, ⋯, n are independent exponential random variables with common parameter equal to (λ₁ + ⋯ + λ_n)/n then there is a monotone coupling of the order statistics X₍₁₎, ⋯, X_(n) and Y₍₁₎, ⋯, Y_(n); that is, it is possible to construct on a common probability space random variables X_i^′, Y_i^′, i = 1, ⋯, n such that for each i, Y_(i)^′ ≤ X_(i)^′ a.s., where the law of the X_i^′ (respectively, the Y_i^′) is the same as the law of the X_i (respectively, the Y_i) 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_i, 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.