Why the mixture failure rate decreases

It is well known that mixtures of decreasing failure rate (DFR) distributions are always decreasing. It turns out that very often mixtures of increasing failure rate (IFR) distributions can decrease as well. In this paper several types of continuous mixtures of IFR distributions are considered. The corresponding limiting behavior of the mixture failure rate function is analyzed for the specific case of mixing which can be interpreted in terms of the proportional hazards model. It is shown that the conditional characteristics (expectation and variance) of the mixing parameter are crucial for the limiting behavior. Several examples are presented and possible generalizations are discussed.