ORACLE INEQUALITIES FOR THE LASSO IN THE COX MODEL.
ORACLE INEQUALITIES FOR THE LASSO IN THE COX MODEL.
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DOI：
10.1214/13-aos1098
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发表时间：
2013-06-01
影响因子：
4.5
通讯作者：
Zhang CH
中科院分区：
文献类型：
--
作者：
Huang J;Sun T;Ying Z;Yu Y;Zhang CH
本文研究了稀疏高维考克斯比例风险回归模型中的绝对惩罚最大偏似然估计，其中时间相关协变量的数目可以大于样本容量。基于Hessian矩阵在真回归系数下的相容性和锥可逆因子的自然扩张，建立了预言不等式。我们的方法也可以证明类似的结果的基础上的限制本征值的扩展。然而，由于相容性和锥可逆因子总是大于相应的限制特征值，所提出的预言不等式更尖锐。在考克斯回归模型中，Hessian矩阵基于删失风险集中的时间相关协变量，因此相容性因子和锥可逆因子以及限制特征值都是随机变量，即使在真实回归系数下对Hessian矩阵进行评估。在温和的条件下，我们证明，这些数量是有界的正常数从下面的时间依赖的协变量，包括协变量的数量是更大的顺序比样本量的情况下。因此，在我们的预言不等式中，相容因子和锥可逆因子可以作为正常数来处理。
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.