APPLICATIONS OF THE HAZARD RATE ORDERING IN RELIABILITY AND ORDER-STATISTICS

The hazard rate ordering is an ordering for random variables which com pares lifetimes with respect to their hazard rate functions. It is str onger than the usual stochastic order for random variables, yet is wea ker than the likelihood ratio ordering. The hazard rate ordering is pa rticularly useful in reliability theory and survival analysis, owing t o the importance of the hazard rate function in these areas. In this p aper earlier work on the hazard rate ordering is reviewed, and extensi ve new results related to coherent systems are derived. Initially we f ix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector (T1, ..., T(n)) to the vector (T1', ..., T (n)'), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is c losed under the formation of series systems, we see that this is not t he caw for parallel systems even when the system is a two-component pa rallel system with exponentially distributed lifetimes. A positive res ult shows that for two-component parallel systems with proportional ha zards (lambda1r(t), lambda2r(t)), the more diverse (lambda1, lambda2) is in the sense of majorization the stronger is the system in the haza rd rate ordering. Unfortunately even this result does not extend to pa rallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering. The principal result of t he paper concerns the hazard rate ordering for the lifetime of a k-out -of-n system. It is shown that if tau(k/n) is the lifetime of a k-out- of-n system, then tau(k/n) is greater than tau(k+1/n) in the hazard ra te ordering for any k. This has an interesting interpretation in the l anguage of order statistics. For independent (not necessarily identica lly distributed) lifetimes T1, ..., T(n), we let T(k:n) represent the kth order statistic (in increasing order). Then it is shown that T(k+1 :n) is greater than T(k:n) in the hazard rate ordering for all k = 1, ..., n - 1. The result does not, however, extend to the stronger likel ihood ratio order.