On usual multivariate stochastic ordering of order statistics from heterogeneous beta variables

Let X i ∼ beta ( α i , 1 ) and Y i ∼ beta ( γ i , 1 ) , i = 1 , 2 , be all independent. We show that ( α 1 , α 2 ) ⪰ m ( γ 1 , γ 2 ) implies ( Y 1 : 2 , Y 2 : 2 ) ≥ s t ( X 1 : 2 , X 2 : 2 ) . We then extend this result to the general case of the proportional reversed hazard rates (PRHR) model.