Reversal of Increasing Failure Rates When Pooling Failure Data

Because mixtures of exponential distributions (with constant failure rate) have the decreasing failure rate (DFR) property, as shown by Proschan in the sixties, it is not unexpected that mixtures of distributions that have a mildly increasing failure rate (IFR) also have this property. What is, perhaps, surprising is that mixtures of IFR distributions with rapidly increasing failure rate may also behave in this manner. Two striking examples of such mixtures are presented here and illustrated graphically. This phenomenon could be the cause of concern, in practice, when it is suspected that a sample is based on pooled data.