THE REVERSED HAZARD RATE-FUNCTION

In this paper we discuss some properties of the reversed hazard rate f unction. 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 h azard rate functions are most useful for lifetimes, and reverse hazard rates are natural if the time scale is reversed. Mixing up these conc epts can often, although not always, lead to anomalies. For example, o ne result gives that if the reversed hazard rate function is increasin g, its interval of support must be (-infinity,b) where b is finite. Co nsequently nonnegative random variables cannot have increasing reverse d hazard rates. Because of this result some existing results in the li terature on the reversed hazard rate ordering require modification. Re versed 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 gi ven that demonstrate this. In studying systems, one problem is to rela te derivatives of hazard rate functions and reversed hazard rate funct ions of systems to similar quantities for components. We give some res ults that address this. Finally, we carry out comparisons for k-out-of -n systems with respect to the reversed hazard rate ordering.