Stochastic comparisons of order statistics from gamma distributions

Let (X1,X2,...,Xn) and (Y1,Y2,...,Yn) be gamma random vectors with common shape parameter [alpha] (0<[alpha][less-than-or-equals, slant]1) and scale parameters ([lambda]1,[lambda]2,...,[lambda]n), ([mu]1,[mu]2,...,[mu]n), respectively. Let X()=(X(1),X(2),...,X(n)), Y()=(Y(1),Y(2),...,Y(n)) be the order statistics of (X1,X2,...,Xn) and (Y1,Y2,...,Yn). Then ([lambda]1,[lambda]2,...,[lambda]n) majorizes ([mu]1,[mu]2,...,[mu]n) implies that X() is stochastically larger than Y(). However if the common shape parameter [alpha]>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results.