| The idea of competing risks in a recurrent event setting is introduced, and in particular, we focus on independent risks. Independent risks occur in many reliability settings dealing with mechanical structures. The main goal of this dissertation is to estimate the cause-specific (or marginal) hazard functions. This leads to estimation of the cumulative hazard function and the survivor function. Two repair strategies are considered. The first repair strategy used is the completely perfect repair strategy in which case all components are replaced upon failure of any one component in a unit. The second repair strategy is the imperfect repair strategy in which case only the failed component is replaced after a failure. We first introduce stochastic process formulations for recurrent competing risks and following the ideas of Jacod [19] develop the likelihood function based on the stochastic processes in order to propose Maximum Likelihood Estimators. Using the results of Borgan [7], it is shown that parametric estimators are consistent and the joint distribution of the parametric estimators is asymptotically multivariate normal. Small sample properties are also explored through simulation. Utilizing doubly-indexed counting processes from Pena, et al. [25] and Sellke [32], a Nelson-Aalen type nonparametric estimator for the cumulative cause-specific hazard function is proposed, which is a step-function. The nonparametric estimator is shown to have bias that decreases geometrically to 0 as n → infinity. It is also shown that the nonparametric estimator, when properly standardized, converges in distribution to a Gaussian process. Simulation studies were also used to demonstrate the finite sample properties of the nonparametric estimator. Examples are also given for both the parametric and nonparametric case for inter-event times that follow a Weibull distribution. |
