STOCHASTIC ORDER OF SAMPLE RANGE FROM HETEROGENEOUS EXPONENTIAL RANDOM VARIABLES

Let X-1, ..., X-n be independent exponential random variables with their respective hazard rates lambda(1), ..., lambda(n), and let Y-1, ..., Y-n be independent exponential random variables with common hazard rate lambda. Denote by X-n:n, Y-n:n and Y-1:n the corresponding maximum and minimum order statistics. X-n:n - X-1:n is proved to be larger than Y-n:n - Y-1:n according to the usual stochastic order if and only if lambda >= ((lambda) over bar (-1)Pi(n)(i=1) lambda(i))(1/(n-1)) with (lambda) over bar = Sigma(n)(i=1) lambda(1/n) . Further, this usual stochastic order is strengthened to the hazard rate order for n = 2. However, a counterexample reveals that this can he strengthened neither to the hazard rate order nor to the reversed hazard rate order in the general case. The main result substantially improves those related ones obtained in Kochar and Rojo [16] and Khaledi and Kochar [13].