ORACLE INEQUALITIES FOR THE LASSO IN THE COX MODEL

• 成果类型: Article
• 署名作者: Huang, Jian; Sun, Tingni; Ying, Zhiliang; Yu, Yi; Zhang, Cun-Hui
• 署名单位: University of Iowa; University of Pennsylvania; Columbia University; Fudan University; Rutgers University System; Rutgers University New Brunswick
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1098
• 发表日期: 2013
• 页码: 1142-1165
• 关键词: PROPORTIONAL HAZARDS MODEL; nonconcave penalized likelihood; variable selection; DANTZIG SELECTOR; REGRESSION-MODEL; LARGE-SAMPLE; sparsity; regularization; Consistency

摘要

We study the absolute penalized maximum partial likelihood estimator in sparse, high-dimensional Cox proportional hazards regression models where the number of time-dependent covariates can be larger than the sample size. We establish oracle inequalities based on natural extensions of the compatibility and cone invertibility factors of the Hessian matrix at the true regression coefficients. Similar results based on an extension of the restricted eigenvalue can be also proved by our method. However, the presented oracle inequalities are sharper since the compatibility and cone invertibility factors are always greater than the corresponding restricted eigenvalue. In the Cox regression model, the Hessian matrix is based on time-dependent covariates in censored risk sets, so that the compatibility and cone invertibility factors, and the restricted eigenvalue as well, are random variables even when they are evaluated for the Hessian at the true regression coefficients. Under mild conditions, we prove that these quantities are bounded from below by positive constants for time-dependent covariates, including cases where the number of covariates is of greater order than the sample size. Consequently, the compatibility and cone invertibility factors can be treated as positive constants in our oracle inequalities.