SOME SUFFICIENT CONDITIONS FOR RELATIVE AGING OF LIFE DISTRIBUTIONS

Abstract

In reliability theory, Cox's proportional hazard model is quite popular and widely used. In many situations, it is observed that failure rates under consideration are not proportional, rather they cross each other. In such situations, an alternative to Cox's proportional hazard model may be monotone hazard ratio model (provided the ratio exists). A notion of relative aging based on increasing hazard ratio was introduced by Kalashnikov and Rachev [19]. Sengupta and Deshpande [40] further explored this model and posited two other notions of relative aging based on increasing reversed failure rate ratio and increasing mean residual life ratio. In this study, for two life distributions, we derive sufficient conditions under which a life distribution ages faster than the other with respect to notions of relative aging described above. These sufficient conditions are easy to verify and can be used in practical applications where one is interested in studying relative aging of two life distributions. Applications of these results to relative aging of weighted distributions have also been illustrated. We also introduce a new relative aging ordering in terms of mean inactivity time order and study its fundamental properties.