On the dependence between the extreme order statistics in the proportional hazards model

Let X1,...,Xn be a random sample from an absolutely continuous distribution with non-negative support, and let Y1,...,Yn be mutually independent lifetimes with proportional hazard rates. Let also X(1)<...<X(n) and Y(1)<...<Y(n) be their associated order statistics. It is shown that the pair (X(1),X(n)) is then more dependent than the pair (Y(1),Y(n)), in the sense of the right-tail increasing ordering of Avérous and Dortet-Bernadet [LTD and RTI dependence orderings, Canad. J. Statist. 28 (2000) 151-157]. Elementary consequences of this fact are highlighted.