The Failure Rates of Mixtures

Abstract

Mixtures of distributions of lifetimes occur in many settings. In engineering applications, it is often the case that populations are heterogeneous, often with a small number of subpopulations. In survival analysis, selection effects can often occur. The concept of a failure rate in these settings becomes a complicated topic, especially when one attempts to interpret the shape as a function of time. Even if the failure rates of the subpopulations of the mixture have simple geometric or parametric forms, the shape of the mixture is often not transparent. Recent results, developed by the author (with Joe, Li, Mi, Savits, and Wond-magegnehu) in a series of papers, are presented. These results focus on general results concerning the asymptotic limit and eventual monotonicity of a mixture, and also the overall behavior for mixtures of specific parametric families. An overall picture is given of different things that influence the behavior of the failure rate of a mixture.