Ordering distributions by scaled order statistics

Motivated by applications in reliability theory, we define a preordering (X₁, ...,X_n) (Y₁ ...,Y_n) of nonnegative random vectors by requiring thek-th order statistic ofa₁X₁,..., a_nX_n to be stochastically smaller than thek-th order statistic ofa₁Y₁, ...,a_nY_n for all choices ofa_i>0,i=1, 2, ...,n. We identify a class of functionsM_{k, n} such that if and only ifE(X)E(Y) for allM_k,n. Some preservation results related to the ordering are obtained. Some applications of the results in reliability theory are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205.