A Note on the Efficiency Based Reliability Measures for Heterogeneous Populations

In many cases, populations in the real world are composed of different subpopulations. Furthermore, in addition to the heterogeneity in the lifetimes of items, there also could be the heterogeneity in the efficiency or performance of items. In this case, the reliability measures should be defined in a different way. In this article, we consider the mixture of stochastically ordered subpopulations. Efficiency based reliability measures are defined when the performance of items in the subpopulations has different levels. Discrete and continuous mixing models are studied. The concept of the association between the lifetime and the performance of items in subpopulations is defined. It is shown that the consideration of efficiency can change the shape of the mixture failure rate dra-matically especially when the lifetime and the performance of items in subpopulations are negatively associated. Furthermore, the modelling method proposed in this paper is applied to the case when the stress levels of the operating environment of items are different