Failure Rates in Heterogeneous Populations

Abstract

Most of the papers on failure rate modeling deal with homogeneous populations. Mixtures of distributions present an effective tool for modeling heterogeneity. In this chapter we consider nonasymptotic and asymptotic properties of mixture failure rates in different settings. After a short introduction, in the first section of this chapter we show (under rather general assumptions) that the mixture failure rate is ‘bent-down’ compared with the corresponding unconditional expectation of the baseline failure rate, which has been proved in the literature for some specific cases. This property is due to an effect where ‘the weakest populations die out first’, explicitly proved mathematically in this section. This should be taken into account when analyzing failure data for heterogeneous populations in practice. 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 the variances when the means are equal. Mixing distributions with equal expectations and different variances can lead to corresponding ordering for mixture failure rates in [0,∞ ) in some specific cases. For a general mixing distribution, however, this ordering is only guaranteed for sufficiently small t. In the second section, the concept of proportional hazards (PH) in a homogeneous population is generalized to a heterogeneous case. For each subpopulation, the PH model is assumed to exist. It is shown that this proportionality is violated for observed (mixture) failure rates. The corresponding bounds for a mixture failure rate are obtained in this case. The change point in the environment is discussed. Shocks – changing the mixing distribution – are also considered. It is shown that shocks with the stochastic properties described also bend down the initial mixture failure rate. Finally, the third section is devoted to new results on the asymptotic behavior of mixture failure rates. The suggested lifetime model generalizes all three conventional survival models (proportional hazards, additive hazards and accelerated life) and makes it possible to derive explicit asymptotic results. Some of the results obtained can be generalized to a wider class of lifetime distributions, but it appears that the class considered is ‘optimal’ in terms of the trade-off between the complexity of a model and the tractability (or applicability) of the results. It is shown that the mixture failure rate asymptotic behavior depends only on the behavior of a mixing distribution near to zero, and not on the whole mixing distribution.