Bathtub functions and burn-in

Components which have bathtub shapes are reasonable candidates for burn-in in the sense that if these components are exposed to standard or elevated operating conditions for a short period of time, they tend to improve. In this paper we study the principle which states that burn-in should occur at or before the point at which a bathtub-shaped failure function starts increasing. To this end we develop three sign change results which characterize bathtub-shaped failure rates. These results are used to show that if a failure rate function is bathtub-shaped, then various other related functions have bathtub or upside-down bathtub shape. Finally, we give a framework for determining when the above principle holds.