Estimating the Reliability of Systems Subject to Imperfect Repair

Abstract

Abstract This study of statistical inference for repairable systems focuses on the development of estimation procedures for the life distribution F of a new system based on data on system lifetimes between consecutive repairs. The Brown—Proschan imperfect-repair model postulates that at failure the system is repaired to a condition as good as new with probability p, and is otherwise repaired to the condition just prior to failure. In treating issues of statistical inference for this model, the article first points out the lack of identifiability of the pair (p, F) as an index of the distribution of interfailure times T 1, T 2, …. It is then shown that data pairs (Ti, Zi ) (i = 1, 2, …) render the parameter pair (p, F) identifiable, where Zi is a Bernoulli variable that records the mode of repair (perfect or imperfect) following the ith failure. Under the assumption that data of the form {(Ti, Zi )} are drawn via inverse sampling until the occurrence of the mth perfect repair, the problem of estimating the parameter pair (p, F) of the Brown—Proschan model is studied. It is demonstrated that the nonparametric maximum likelihood estimator of F exists only in special cases, but that a neighborhood maximum likelihood estimator [Fcirc] (using the language of Kiefer and Wolfowitz 1956) always exists and may be derived in closed form. Under mild assumptions, the strong uniform consistency of [Fcirc] is demonstrated, as is the weak convergence of an appropriately scaled version of [Fcirc] to a Gaussian process. It is noted that these results apply to other experimental designs, such as renewal testing, and that they can be extended to the age-dependent imperfect-repair model of Block, Borges, and Savits (1985).