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This paper shows that: (1) the mean residual life (MRL) of a component has an upside-down bathtub-shape if the component has a bathtub-shape failure-rate function, but the converse does not hold; and (2) there is an optimal burn-in policy to maximize the MRL when the underlying lifetime distribution has a bathtub-shape failure rate.>
Jie Mi (Sun,) studied this question.