LIKELIHOOD RATIO AND MEAN RESIDUAL LIFE ORDERS FOR ORDER STATISTICS OF HETEROGENEOUS RANDOM VARIABLES

Abstract

Let X1:n ≤ X2:n ≤ ··· ≤ Xn:n denote the order statistics of a set of independent and not necessarily identically distributed random variables X1,..., Xn. Under mild assumptions, it is shown that Xk−1:n−1 ≤lrXk:n for k = 2,..., n if X1 ≤lrX2 ≤lr ··· ≤lrXn and that Xk:n ≤lrXk:n−1 for k = 1,..., n − 1 if X1 ≥lrX2 ≥lr ··· ≥lrXn, where ≤lr denotes the likelihood ratio order. Concerning the mean residual life order (≤mrl), it is shown that Xn−1:n−1 ≤mrlXn:n if Xj ≤mrlXn for j = 1,..., n − 1. Two counterexamples are also given to illustrate that Xk−1:n−1 ≤mrlXk:n in this case is, in general, not true for k < n.