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.

Relative aging of (n − k + 1)‐out‐of‐n systems based on cumulative hazard and cumulative reversed hazard functions

“…Firstly, we introduce some partial orders that describe relative aging. To comprehend them in more detail, reader may refer to Di Crescenzo (2000), Di Crescenzo and Longobardi (2015), Ding, Fang, and Zhao (2017), Ding and Zhang (2018), Kalashnikov and Rachev (1986), Kayid et al (2017), Lai and Xie (2003), Li and Li (2016), Misra and Francis (2015), Misra et al (2017), Rezaei et al (2015), Rowell and Siegrist (1998), and Sengupta and Deshpande (1994).…”

“…This notion of relative aging is weaker than that based on ratio of failure rates. Recently, more notions of relative aging based on monotonicity of ratios of other reliability measures (such as mean residual life function and mean inactivity time function) are introduced in the literatures (Finkelstein, 2006;Kayid, Izadkhah, & Zuo, 2017;Misra, Francis, & Naqvi, 2017;Rezaei, Gholizadeh, & Izadkhah, 2015). For some other contributions on relative aging of life distributions, readers may refer to Di Crescenzo and Longobardi (2015), Fernandez-Ponce et al (1998), Kochar (1989), Parsa et al (2018), and Rowell and Siegrist (1998).…”

Results on Relative Mean Residual Life and Relative Cumulative Residual Entropy

“…As Misra et al [13] have mentioned, since increasing mean residual life describes negative aging, we believe that the correct way of defining relative aging in terms of mean residual function is as given in Definition 2 (iv).…”

“…Before proceeding to give the main results of the paper, we overview some preliminary concepts of ageing and standard partial orderings describing relative ageing. (For more details of these concepts see, for example, [18], [6], [19], [10], [13]).…”

“…Li and Li [12] have studied the relative ageing order on series and parallel systems with mutually statistically independent and heterogeneous component lifetimes. Misra et al [13] have introduced a new notion of relative aging based on the ratio of the mean inactivity time functions.…”