HAZARD RATE ORDERING OF K-OUT-OF-N SYSTEMS

Let (X(1),X(2),...,X(n)) be a set of independent continuous random lifetimes and let {Y1, Y-2,...,Y-n} be another set of independent continuous random lifetimes. Let X((1))less than or equal to X((2)) less than or equal to...less than or equal to X((n)) be the order statistics associated with the X(i)'s, and let Y-(1)less than or equal to Y-(2)less than or equal to...less than or equal to Y-(n) be the order statistics associated with the Y-i's. Several recent papers have given conditions under which X((k))less than or equal to(hr)Y((k)),k=1,2,...,n, where 'less than or equal to(hr),' denotes the hazard rate order. For example, it is known that if the X(i)'s are independent and identically distributed, and if the Y-i's are independent and identically distributed, and if X(i) less than or equal to(hr)Y(i),i=1,2,...n, then X((k)) less than or equal to(hr)Y((k)). In fact, if the X(i)'s are not necessarily identically distributed and the Y-i's are not necessarily identically distributed, but X(alpha)less than or equal to(hr)Y(beta),alpha,beta=1,2,...,n, then it is still true that X((k))less than or equal to(hr)Y((k)),k=1,2,...,n. The purpose of this paper is to obtain an even stronger result. Our proof is different than the other proofs in the literature and is more intuitive. Some applications in reliability theory are given.