Oracle inequalities for the lasso in the Cox model
Cox模型中Lasso的预言机不等式AI 解读 ↓
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Cox模型中Lasso的预言机不等式
- 研究问题
- 在含时变协变量、维度可超过样本量的稀疏Cox模型中，如何为Lasso建立误差保证并控制删失风险集带来的随机矩阵性质？
- 方法与思路
- 作者研究绝对值惩罚的最大部分似然估计，扩展真实系数处Hessian的兼容性、锥可逆性及受限特征值概念。
- 材料陈述的贡献
- 摘要称建立较受限特征值版本更紧的预言机不等式，并在所述温和条件下证明相关随机因子有正的常数下界。
- 阅读关注点
- 应核对删失与时变协变量条件、惩罚参数选择、随机因子下界的概率保证，以及不等式比较的具体误差标准。
- 阅读边界
- 索引摘要未注明稿件版本，也未列出所谓温和条件和界的形式；更紧的不等式不直接保证每个样本中的预测或估计优势。
原文摘要
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.