Nonparametric statistical inference for survival data

Abstract

Censored data, one of the most common data types, arise frequently in many fields of modern science, e.g., health science, reliability, economics, finance, etc. The most prominent feature of this kind of data is that the occurrence of the event could not be observed exactly. Right censored data and interval censored data are among the most popular ones. Over the past decades, there have been numerous state-of-the-art methodolo-gies in survival analysis literature to handle censoring. This thesis would focus on the nonparametric statistical inference of right censored data and interval censored data. As the first part of this thesis, a penalized nonparametric maximum likelihood estimation of the log-hazard function is introduced in analyzing the right censored data. The smoothing spline is employed for a smooth estimation. The most appealing fact is that a functional Bahadur representation is established, which serves as a key step for nonparametric inference of the unknown parameter/function. Asymptotic properties of the resulting estimate of the unknown log-hazard function are proved. Furthermore, the local confidence interval and simultaneous confidence band of the unknown log-hazard function are provided, along with a local and global likelihood ratio tests. We also investigate issues related to the asymptotic efficiency. As the second part of this thesis, the aforementioned nonparametric inference approach is extended to handle interval censored data. In particular, we focus on the nonparametric inference of the cumulative hazard function, instead of the log-hazard function of the interval censored data. Similarly, we have derived a functional Bahadur representation and established the asymptotic properties of the resulting estimate of the cumulative function. Particularly, the global asymptotic properties are justified under regularity conditions. A likelihood ratio test is also provided. To the best of our knowledge, there is no report in the literature on the asymptotic properties of a smoothing spline-based nonparametric estimate for the interval censored data. The theoretical results are validated by extensive simulation studies. Applications are illustrated with some real datasets. A few discussions and closing remarks are given.