On an inverse problem in mixture failure rates modelling

Mixtures of decreasing failure rate (DFR) distributions are always DFR. It turns out that very often mixtures of increasing failure rate distributions can decrease or show even more complicated patterns of dependence on time. For studying this and other relevant effects two simple models of mixing with additive and multiplicative failure rates are considered. It is shown that for these models an inverse problem can be solved, which means that given an arbitrary shape of the mixture failure rate and a mixing distribution, the failure rate for a governing distribution can be uniquely obtained. Some examples are considered where this operation can be performed explicitly. Possible generalizations are discussed. Copyright (C) 2001 John Wiley & Sons, Ltd.