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

Let Xi∼beta(αi,1) and Yi∼beta(γi,1), i=1,2i=1,2, be all independent. We show that (α1,α2)⪰m(γ1,γ2) implies (Y1:2,Y2:2)≥st(X1:2,X2:2)(Y1:2,Y2:2)≥st(X1:2,X2:2). We then extend this result to the general case of the proportional reversed hazard rates (PRHR) model.