Understanding the Shape of the Mixture Failure Rate (with Engineering and Demographic Applications)

Mixtures of distributions are usually effectively used for modeling heterogeneity. It is well known that mixtures of DFR distributions are always DFR. On the other hand, mixtures of IFR distributions can decrease, at least in some intervals of time. As IFR distributions often model lifetimes governed by ageing processes, the operation of mixing can dramatically change the pattern of ageing. Therefore, the study of the shape of the observed (mixture) failure rate in a heterogeneous setting is important in many applications. We study discrete and continuous mixtures, obtain conditions for the mixture failure rate to tend to the failure rate of the strongest populations and describe asymptotic behavior as ∞ → t . Some demographic and engineering examples are considered. The corresponding inverse problem is discussed.