Efficient semiparametric estimation of time-censored intensity-reduction models for repairable systems

The rate reduction models have been widely used to model the recurrent failure data for their capabilities in quantifying the repair effects. Despite the widespread popularity, there have been limited studies on statistical inference of most failure rate reduction models. In view of this fact, this study proposes a semiparametric estimation framework for a general class of such models, called extended geometric failure rate reduction (EGFRR) models. Covariates are considered in our analysis and their effects are modeled as a log-linear factor on the baseline failure rate. Unlike the existing inference methods for the EGFRR models that assume the failure data are censored at a fixed number of failures, our study considers covariates and time-censoring, which are more common in practice. The semiparametric maximum likelihood (ML) estimators are obtained by carefully constructing the likelihood function. Asymptotic properties including consistency and weak convergence of the ML estimators are established by using the properties of the martingale process. In addition, we show that the semiparametric estimators are asymptotically efficient. A real example from the automobile industry illustrates the usefulness of the proposed framework and extensive simulations show its outstanding performance when comparing with the existing methods.