Random wear models in reliability theory

Abstract

Gaver (1963) and Antelman and Savage (1965) have proposed models for the distribution of the time to failure of a simple device exposed to a randomly varying environment. Each model represents cumulative wear as a specified function of a non-negative stochastic process with independent increments, and assumes that the reliability of the device is conditioned upon realizations of this process. From these models are derived the corresponding unconditional joint distributions for the random failure time vector of n independent, identical devices exposed to the same realization of the wear process. It is shown that the identical failure time distribution for one component can arise from each model. In the Gaver model simultaneous failure times occur with positive probability. The probabilities of specific tie configurations are developed. For an interesting class of Gaver models involving a time scale parameter, the maximum likelihood estimates from several devices in one environment are examined. In that case the tie configuration probability does not depend on the parameter. For the corresponding Antelman-Savage models a consistent sequence of estimators is obtained; the maximum likelihood theory did not appear tractable.