ORACLE INEQUALITIES FOR THE LASSO IN THE COX MODEL

Jian Huang; Tingni Sun; Zhiliang Ying; Yi Yu; Cun-Hui Zhang

doi:10.1214/13-AOS1098

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ORACLE INEQUALITIES FOR THE LASSO IN THE COX MODEL

Journal article

Open access

Peer reviewed

ORACLE INEQUALITIES FOR THE LASSO IN THE COX MODEL

Jian Huang, Tingni Sun, Zhiliang Ying, Yi Yu and Cun-Hui Zhang

Annals of statistics, Vol.41(3), pp.1142-1165

06/01/2013

DOI: 10.1214/13-AOS1098

PMCID: PMC3786146

PMID: 24086091

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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.

absolute penalty

Proportional hazards

survival analysis

regression

regularization

oracle inequality

Details

Title: Subtitle: ORACLE INEQUALITIES FOR THE LASSO IN THE COX MODEL
Creators: Jian Huang - Rutgers University
Tingni Sun - Rutgers University
Zhiliang Ying - Rutgers University
Yi Yu - Rutgers University
Cun-Hui Zhang - Rutgers University
Resource Type: Journal article
Publication Details: Annals of statistics, Vol.41(3), pp.1142-1165
DOI: 10.1214/13-AOS1098
PMID: 24086091
PMCID: PMC3786146
ISSN: 0090-5364
eISSN: 2168-8966
Grant note: R37 GM047845 || GM / National Institute of General Medical Sciences : NIGMS
Language: English
Date published: 06/01/2013
Academic Unit: Statistics and Actuarial Science
Record Identifier: 9983985803002771

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