On Mixture Failure Rates Ordering

Mixtures of increasing failure rate distributions (IFR) can decrease at least in some intervals of time. Usually, this property can be observed asymptotically as t → ∞. This is due to the fact that the mixture failure rate is “bent down” compared with the corresponding unconditional expectation of the baseline failure rate, which was proved previously for some specific cases. We generalize this result and discuss the “weakest populations are dying first” property, which leads to the change in the failure rate shape. We also consider the problem of mixture failure rate ordering for the ordered mixing distributions. Two types of stochastic ordering are analyzed: ordering in the likelihood ratio sense and ordering in variances when the means are equal.