Inferences on a life distribution by sampling from the ages or the ages at death

Consider a system where units having independent and identically distributed lifetimes enter according to a nonhomogeneous Poisson process. After the unit's life in the system, the unit departs the system. For a fixed system time, this paper relates the units' common underlying life distribution with the distribution of the ages of units in the system, the distribution for the system life of units that departed the system and the distribution for the system life of units that have recently departed the system. Results can be used to estimate the underlying life distribution or a truncated version of that distribution based on the ages and/or most recent ages at death in both one sample and two sample situations. Results include a complete characterization of the possible distribution of the ages of those units in the system, how to estimate the underlying life distribution from the most recent ages at death, and how to test for an underlying monotone failure rate function based on independent samples from the ages and most recent ages at death. Two sample inferences that involve a likelihood ratio ordering make use of the results in Dykstra et al. (1995, J Amer Statisc Assoc 90(431):1030-1040), which provides the maximum likelihood estimators and a likelihood ratio test when the two distributions satisfy a likelihood ratio ordering. For the ages of the active units and the ages at death among the departed units, limits for their distributions and strong limiting results for their empirical distributions will be provided.