On the range of heterogeneous samples

Abstract

Let R n be the range of a random sample X 1 , … , X n of exponential random variables with hazard rate λ . Let S n be the range of another collection Y 1 , … , Y n of mutually independent exponential random variables with hazard rates λ 1 , … , λ n whose average is λ . Finally, let r and s denote the reversed hazard rates of R n and S n , respectively. It is shown here that the mapping t ↦ s ( t ) / r ( t ) is increasing on ( 0 , ∞ ) and that as a result, R n = X ( n ) − X ( 1 ) is smaller than S n = Y ( n ) − Y ( 1 ) in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X ( n ) is seen to be more stochastically increasing in X ( 1 ) than Y ( n ) is in Y ( 1 ) . In other words, the pair ( X ( 1 ) , X ( n ) ) is more dependent than the pair ( Y ( 1 ) , Y ( n ) ) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X 1 , … , X n form a random sample from a continuous distribution while Y 1 , … , Y n are mutually independent lifetimes with proportional hazard rates.