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

Let X 1 , … , X n be a random sample from an absolutely continuous distribution with non-negative support, and let Y 1 , … , Y n 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.