Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability (original) (raw)

Let ðX 1 ; X 2 ; y; X n Þ and ðY 1 ; Y 2 ; y; Y n Þ be gamma random vectors with common shape parameter a ð0oap1Þ and scale parameters ðl 1 ; l 2 ; y; l n Þ; ðm 1 ; m 2 ; y; m n Þ; respectively. Let X ðÞ ¼ ðX ð1Þ ; X ð2Þ ; y; X ðnÞ Þ; Y ðÞ ¼ ðY ð1Þ ; Y ð2Þ ; y; Y ðnÞ Þ be the order statistics of ðX 1 ; X 2 ; y; X n Þ and ðY 1 ; Y 2 ; y; Y n Þ: Then ðl 1 ; l 2 ; y; l n Þ majorizes ðm 1 ; m 2 ; y; m n Þ implies that X ðÞ is stochastically larger than Y ðÞ : However if the common shape parameter a41; 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.