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June 2013 Oracle inequalities for the lasso in the Cox model
Jian Huang, Tingni Sun, Zhiliang Ying, Yi Yu, Cun-Hui Zhang
Ann. Statist. 41(3): 1142-1165 (June 2013). DOI: 10.1214/13-AOS1098

Abstract

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.

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Jian Huang. Tingni Sun. Zhiliang Ying. Yi Yu. Cun-Hui Zhang. "Oracle inequalities for the lasso in the Cox model." Ann. Statist. 41 (3) 1142 - 1165, June 2013. https://doi.org/10.1214/13-AOS1098

Information

Published: June 2013
First available in Project Euclid: 13 June 2013

zbMATH: 1292.62135
MathSciNet: MR3113806
Digital Object Identifier: 10.1214/13-AOS1098

Subjects:
Primary: 62N02
Secondary: 62G05

Keywords: absolute penalty , Oracle inequality , proportional hazards , regression , regularization , Survival analysis

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3 • June 2013
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