Pasquale Sullo

Dept. of Decision Sciences and Engineering Systems

Rensselaer Polytechnic Institute

Troy, New York 12180

Continuous mixtures of life distributions have long been used to model heterogeneous populations in reliability, survival, and other durational event studies. It is well known that such mixtures may have properties that are radically different than those of the families of distributions being mixed. This presentation will concentrate on gamma mixtures of families of strictly increasing, proportional hazard rate distributions, in which the baseline hazard rate is a Polya frequency function of order 2 (PF₂). PF₂ functions represent a large class with many applications in probability and statistics. However, nothing seems to have been written concerning PF₂ hazard rates. Yet, all the standard increasing hazard rate (IHR) distributions (and many more) have PF₂ hazard rates. In the context of gamma mixtures of increasing, proportional hazard rate distributions, it is shown that the hazard rate of the resulting distribution can be decreasing, increasing to a maximum and then decreasing, or increasing (IHR). For most reliability distributions, e.g., Weibull, gamma, truncated normal, truncated logistic, the second case holds. However, when the last case holds the hazard rate can only increase asymptotically to a finite constant. Sufficient conditions for these three cases will be given.