ON STOCHASTIC ORDERS FOR THE LIFETIME OF A k-OUT-OF-n SYSTEM

Abstract

Let τ_k|n denote the lifetime of a k-out-of-n system, where the n components have independent lifetimes T_i with completely arbitrary distribution F_i, i = 1,..., n. It is shown that τ_k+1|n ≤_hr τ_k|n, τ_k|n ≤_hr τ_k−1|n−1, and τ_k|n−1 ≤_hr τ_k|n if T_i ≤_hrT_n, i = 1,..., n − 1; τ_k+1|n ≤_rh τ_k|n, τ_k−1|n ≤_rh τ_k|n, and τ_k|n ≤_rh τ_k−1|n−1 if T_n ≤_rhT_i, i = 1,..., n − 1. These results are available in the literature for the special case of F_i's being absolutely continuous. Also, even in this case, the proofs are often tedious and use the concept of “totally positive of order infinity in differences of k.” In contrast, the proofs given here are simple and elegant and do not use the above concept.