Stochastic comparisons of order statistics from gamma distributions

Abstract

Let ( X 1, X 2,…, X n ) and ( Y 1, Y 2,…, Y n ) be gamma random vectors with common shape parameter α (0<α⩽1) and scale parameters ( λ 1, λ 2,…, λ n ), ( μ 1, μ 2,…, μ n ), respectively. Let X ()=( X (1), X (2),…, X ( n) ), Y ()=( Y (1), Y (2),…, Y ( n) ) be the order statistics of ( X 1, X 2,…, X n ) and ( Y 1, Y 2,…, Y n ). Then ( λ 1, λ 2,…, λ n ) majorizes ( μ 1, μ 2,…, μ n ) 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.