Mean residual life and its association with failure rate

The life time distributions having decreasing, increasing, or upside-down bathtub shaped MRL (mean residual life) are used as models in many applications. Mi (1995) has shown that if a component has a bathtub shape failure rate function, then the MRL is unimodal but the converse does not hold. This paper develops sufficient conditions for the unimodal MRL to imply that the failure rate function has a bathtub shape