Semiparametric Inference for the Functional Cox Model

This article studies penalized semiparametric maximum partial likelihood estimation and hypothesis testing for the functional Cox model in analyzing right-censored data with both functional and scalar predictors. Deriving the asymptotic joint distribution of finite-dimensional and infinite-dimensional estimators is a very challenging theoretical problem due to the complexity of semiparametric models. For the problem, we construct the Sobolev space equipped with a special inner product and discover a new joint Bahadur representation of estimators of the unknown slope function and coefficients. Using this key tool, we establish the asymptotic joint normality of the proposed estimators and the weak convergence of the estimated slope function, and then construct local and global confidence intervals for an unknown slope function. Furthermore, we study a penalized partial likelihood ratio test, show that the test statistic enjoys the Wilks phenomenon, and also verify the optimality of the test. The theoretical results are examined through simulation studies, and a right-censored data example from the Improving Care of Acute Lung Injury Patients study is provided for illustration. Supplementary materials for this article are available online.

本文针对同时包含函数型预测变量与标量预测变量的右删失数据，研究函数型Cox模型（functional Cox model）的惩罚半参数极大偏似然估计与假设检验问题。由于半参数模型的复杂性，推导有限维估计量与无限维估计量的渐近联合分布是极具挑战性的理论难题。针对该问题，本文构建了配备特殊内积的索伯列夫空间（Sobolev space），并发现未知斜率函数与系数估计量的新型联合巴哈杜尔表示（Bahadur representation）。依托这一关键工具，本文确立了所提估计量的渐近联合正态性与估计斜率函数的弱收敛性质，进而构建了未知斜率函数的局部与全局置信区间。此外，本文还研究了惩罚偏似然比检验，证明该检验统计量满足威尔克斯现象（Wilks phenomenon），并验证了该检验的最优性。本文通过模拟研究对理论结果进行了验证，并辅以来自急性肺损伤患者诊疗质量改善研究的右删失数据实例展开说明。本文的补充材料可在线获取。