‘Understanding the shape of the mixture failure rate’ by Maxim Finkelstein: Discussion 2

Abstract

DISCUSSIONFirst of all I would like to say that it is both an honor and a pleasure to join Professor Finkelstein in the discussion of this very interesting and relevant topic.The 'strange' behaviour of the failure (or hazard) rate function of heterogeneous populations has been a key point in the development of Reliability and Survival theories. The failure rate of classical distribution models are usually increasing or decreasing. However, usually, the failure rate functions obtained from real data sets are not monotone. This fact can be explained by a mixture of classical distribution models, that is, by the heterogeneity of the population. The mixture of two decreasing failure rate (DFR) models is also DFR, but the failure rate of a mixture of two increasing failure rate (IFR) models (i.e. models that have ageing) can have several shapes including a decreasing behaviour in some intervals. A well-known example is the case of two exponential distributions that have constant failure rates, but their mixture is strictly DFR. The shape is more complicated if we consider general discrete or continuous mixtures.The paper by Finkelstein shows that this strange behaviour is a simple consequence of the mixture (heterogeneity) and the fact that the weakest populations are dying first (the WPDF principle). Therefore, in the two-component mixture case (Section 2), if the failure rate functions are ordered, then the mixture failure rate moves from one failure rate to the other and, finally (when the time increases), it is close to the failure rate of the strongest population. This fact explains the behaviour of the failure rate of the mixture of two exponential distributions. A similar interpretation is given