Motivated by applications in reliability theory, we define a preordering \((X_ 1,...,X_ n)\lesssim^{(k)}(Y_ 1,...,Y_ n)\) of nonnegative random vectors by requiring the k-th order statistic of \(a_ 1X_ 1,...,a_ nX_ n\) to be stochastically smaller than the k-th order statistic of \(a_ 1Y_ 1,...,a_ nY_ n\) for all choices of \(a_ i>0\), \(i=1,2,...,n\). We identify a class of functions \(M_{k,n}\) such that \(X\lesssim^{(k)}Y\) if and only if \(E\phi\) (X)\(\leq E\phi (Y)\) for all \(\phi \in M_{k,n}\). Some preservation results related to the ordering \(\lesssim^{(k)}\) are obtained. Some applications of the results in reliability theory are given.