Stochastic comparisons of order statistics from gamma distributions

Let (X1,X2,…,Xn) and (Y1,Y2,…,Yn) be gamma random vectors with common shape parameter α (0<α⩽1) and scale parameters (λ1,λ2,…,λn), (μ1,μ2,…,μn), 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 (λ1,λ2,…,λn) majorizes (μ1,μ2,…,μn) implies that X() is stochastically larger than Y(). However if the common shape parameter α>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.