Bathtub and Related Failure Rate Characterizations

Sufficient conditions are obtained that ensure that a lifetime density has a bathtub-shaped failure rate. Analogous conditions handle increasing, decreasing, and upside-down bathtub-shaped failure rates. Application of these results to exponential families of densities is particularly straightforward and effective. Examples are furnished that introduce new bathtub models and illustrate the use of the general results for existing models. Examples involving mixtures are also considered.