Semiparametric statistical inference for functional survival models

Abstract

This thesis focuses on the development of semiparametric inference for the functional Cox proportional hazards model and the functional additive hazards model with right-censored data. We propose a penalized partial likelihood approach and a penalized pseudo-score function approach to the estimation of the model parameters of the functional Cox proportional hazards model and that of the functional additive hazards model, respectively. We establish asymptotic properties which include the consistency, the convergence rate, and the limiting distribution of the proposed estimators. To this end, we investigate the joint Bahadur representation of finite-dimensional and infinite-dimensional estimators in the Sobolev space with proper inner products. One major contribution made to the study of the functional Cox proportional hazards model and the functional additive hazards model is that the asymptotic joint normality of the estimators of the functional coefficient and the scalar coefficient is derived. Furthermore, the partial likelihood ratio test is developed and is shown to be optimal under the functional Cox proportional hazards model. These two important issues are not addressed in the previous research. Our new results provide more insights and deeper understanding about the effects of functional predictors on the hazard function. The theoretical results are validated by simulation studies, and the applications of the proposed models are illustrated with a real dataset. Some discussions and closing remarks are given.