INEQUALITIES FOR DISTRIBUTIONS WITH INCREASING FAILURE RATE

Abstract

This chapter reviews inequalities for distributions with increasing failure rate (IFR). The IFR distributions have been widely studied. A distribution on [0,∞) is defined to be IFR if the residual life is stochastically decreasing in t, that is, ▪ is decreasing in t for each s > 0. Analogues of the results of Brown for the renewal theory for DFR interarrival times do not hold in the case of IFR. The IFR class is more difficult to penetrate than DFR for the properties of interest. In some cases, no close analogue of the DFR result holds; in others, the IFR analogue is weaker. IFR on [0,∞) are absolutely continuous except for an atom at the right-hand endpoint of the support. The chapter also discusses renewal function inequalities and exponential approximations.