On sufficient conditions for mean residual life and related orders

A well known sufficient condition for the mean residual life order of two random variables is the hazard rate order of the two random variables. The hazard rate order is characterized by the monotonicity of the ratio of the two survival functions. However in many cases this ratio is non monotone and the hazard rate order does not hold. The purpose is to show that, in some situations, this non monotonicity is still a sufficient condition for the mean residual life order, under some additional mild conditions. Applications to compare some parametric models of distributions and generalized order statistics are provided. Similar results are given for the mean inactivity time order.