Some Nonparametric Statistical Inference In Survival Analysis

We consider two issues in this dissertation,the estimation of hazard level set and its asymptotic properties,and the local asymptotic normality of the nonpara-metric estimation for the Cox model.Firstly,the hazard rate function plays a fundamental role in survival analysis.Its statistical inference methods have been systemically and exclusively studied.When does the hazard rate reach a particular warning level and how to construct the corresponding confidence region?This is a basic question of interest to the investigator but largely left to be explored in practice.We define a level set of hazard rate to address this issue and propose a kernel smoothing estimator for such a level set.In terms of the Hausdorff distance,we establish the consistency,convergence rate and asymptotic distribution of the level set estimator.The validity of the proposed confidence set,based on the bootstrap method,for the level set of hazard rate function is theoretically justified.We conduct comprehensive simulation studies to assess the finite-sample performance of the proposed method,which is further illustrated with a breast cancer study.Secondly,the penalized nonparametric Cox model,has been widely used in many fields,was a simple and effective method.Based on penalized partial like-lihood,however,the limiting distribution of the nonparametric estimation,was crucial in statistical inference for the relative risk function,has never been studied in detail.Under some regularity conditions,we proved the local asymptotic nor-mality of the corresponding estimation by using reproducing kernel Hilbert space,empirical process theory,and functional Bahadur representation.Furthermore,we suggested to construct the corresponding local confidence intervals by bootstrap method.Some simulation studies and Stanford heart transplant study are con-ducted to validate the proposed method and compare with Bayesian confidence intervals.