Pj. Boland et al., APPLICATIONS OF THE HAZARD RATE ORDERING IN RELIABILITY AND ORDER-STATISTICS, Journal of Applied Probability, 31(1), 1994, pp. 180-192
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.