Properties of residual mixing distributions resulting from arbitrary mixtures of exponential life distributions

Abstract

A mixture of failure rates can be present in an apparently homogeneous population of "devices" due to variability either in their manufacture or in the severity of their service environments. An initial mixing distribution is the probability distribution for different failure rates in such a population. This distribution may be updated to yield its related residual mixing distribution, which is the probability distribution for different failures rates in the population of survivors 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. Properties of expected value ordering, stochastic ordering, failure rate ordering and likelihood ration ordering are investigated to show that, of these, only likelihood ratio ordering between the initial mixing distributions is sufficient to guarantee an ordering between the expected values of their respective residual mixing distributions over time.