Stochastic ordering in residual mixing distributions

Abstract

A mixture of failure rates can be present in an apparently homogeneous population of "devices" because of variability either in their manufacture or in the severity of their service environments. A mixing distribution is the probability distribution for different failure rates in such a population. A residual mixing distribution is the probability distribution for different failure rates in the population of surviving devices after a specified period of service or "burn in." Residual mixing distributions resulting from arbitrary mixtures of constant failure rates are shown to be stochastically ordered (decreasingly) as the period of service or burn in is increased, and to approach in the limit a distribution degenerate at the smallest failure rate "present" in the population. The results are direct applications of basic propositions concerning monotonicity properties of residual mixing random variables and their expectations. They contain some well known results about the failure rate of an item drawn from a mixed population of constant failure rates as immediate consequences. (Author)