Modeling lifetimes with unknown initial age

The failure rate of a distribution, which describes a lifetime random variable with unknown initial age, is studied. A specific model of mixing formalizes this setting. In accordance with this model, it turns out that under certain assumptions the operation of mixing distributions with increasing failure rates can result in the mixture failure rate with a bathtub shape: decreasing in the initial interval of time and then increasing, asymptotically approaching the failure rate of the baseline distribution. Two specific cases of the baseline distribution are considered: the Weibull and the truncated extreme value distributions. A possibility of mixing via the mean residual lifetime function is discussed.