Inequalities for ordered sums

Summary Letx _i =y _i +z _i ,i=1, ...n, and writex ₍₁₎≦...≦x _(n) , with corresponding notation for the orderedy _i andz _i . It is shown, for example, that ,r=1, ...n. Inequalities are also obtained for convex (or concave) functions of thex _(i) . The results lead immediately to bounds for the expected values of order statistics in nonstandard situations in terms of simpler expectations. A small numerical example illustrates the method. Research supported by U.S. Army Research Office.