The Reversed Hazard Rate Function

Abstract

In this paper we discuss some properties of the reversed hazard rate function. This function has been shown to be useful in the analysis of data in the presence of left censored observations. It is also natural in discussing lifetimes with reversed time scale. In fact, ordinary hazard rate functions are most useful for lifetimes, and reverse hazard rates are natural if the time scale is reversed. Mixing up these concepts can often, although not always, lead to anomalies. For example, one result gives that if the reversed hazard rate function is increasing, its interval of support must be (—∞, b) where b is finite. Consequently nonnegative random variables cannot have increasing reversed hazard rates. Because of this result some existing results in the literature on the reversed hazard rate ordering require modification.Reversed hazard rates are also important in the study of systems. Hazard rates have an affinity to series systems; reversed hazard rates seem more appropriate for studying parallel systems. Several results are given that demonstrate this. In studying systems, one problem is to relate derivatives of hazard rate functions and reversed hazard rate functions of systems to similar quantities for components. We give some results that address this. Finally, we carry out comparisons for k-out-of-n systems with respect to the reversed hazard rate ordering.