Semiparametric estimate of the efficiency of imperfect maintenance actions for a gamma deteriorating system

A system is considered, which is deteriorating over time according to a non homogeneous gamma process with unknown parameters. The system is subject to periodic and instantaneous imperfect maintenance actions (repairs). Each imperfect repair removes a proportion rho of the accumulated degradation since the previous repair. The parameter rho hence appears as a measure for the maintenance efficiency. This model is called arithmetic reduction of degradation of order 1. The system is inspected right before each maintenance action, thus providing some multivariate measurement of the successively observed deterioration levels. Based on these data, a semiparametric estimator of rho is proposed, considering the parameters of the underlying gamma process as nuisance parameters. This estimator is mainly based on the range of admissible rho's, which depends on the data. Under technical assumptions, consistency results are obtained, with surprisingly high convergence rates (up to exponential). The case where several i.i.d. systems are observed is next envisioned. Consistency results are obtained for the efficiency estimator, as the number of systems tends to infinity, with a convergence rate that can be higher or lower than the classical square root rate. Finally, the performances of the estimators are illustrated on a few numerical examples. (C) 2019 Elsevier B.V. All rights reserved.