On the range of heterogeneous samples

Let RnRn be the range of a random sample X1,…,XnX1,…,Xn of exponential random variables with hazard rate λλ. Let SnSn be the range of another collection Y1,…,YnY1,…,Yn of mutually independent exponential random variables with hazard rates λ1,…,λnλ1,…,λn whose average is λλ. Finally, let rr and ss denote the reversed hazard rates of RnRn and SnSn, respectively. It is shown here that the mapping t↦s(t)/r(t)t↦s(t)/r(t) is increasing on (0,∞)(0,∞) and that as a result, Rn=X(n)−X(1)Rn=X(n)−X(1) is smaller than Sn=Y(n)−Y(1)Sn=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)X(n) is seen to be more stochastically increasing in X(1)X(1) than Y(n)Y(n) is in Y(1)Y(1). In other words, the pair (X(1),X(n))(X(1),X(n)) is more dependent than the pair (Y(1),Y(n))(Y(1),Y(n)) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X1,…,XnX1,…,Xn form a random sample from a continuous distribution while Y1,…,YnY1,…,Yn are mutually independent lifetimes with proportional hazard rates.