Modeling a failure rate for a mixture of distribution functions

It is well known that mixtures of decreasing failure rate (DFR) distributions are always DFR. it turns out that, very often, mixtures of increasing failure rate (IFR) distributions can decrease at least in some intervals of time. Usually, this property can be observed asymptotically as t --> infinity. In this article, several types of underlying continuous IFR distribution are considered. Two models of mixing are studied: additive and multiplicative. The limiting behavior of a mixture failure rate function is analyzed. 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.