Non-Asymptotic Oracle Inequalities for the High-Dimensional Cox Regression via Lasso.

We consider finite sample properties of the regularized high-dimensional Cox regression via lasso. Existing literature focuses on linear models or generalized linear models with Lipschitz loss functions, where the empirical risk functions are the summations of independent and identically distributed (iid) losses. The summands in the negative log partial likelihood function for censored survival data, however, are neither iid nor Lipschitz.We first approximate the negative log partial likelihood function by a sum of iid non-Lipschitz terms, then derive the non-asymptotic oracle inequalities for the lasso penalized Cox regression using pointwise arguments to tackle the difficulties caused by lacking iid Lipschitz losses.

[1]  David R. Cox,et al.  Regression models and life tables (with discussion , 1972 .

[2]  R. Gill,et al.  Cox's regression model for counting processes: a large sample study : (preprint) , 1982 .

[3]  P. Massart.  The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality , 1990 .

[4]  M. Talagrand,et al.  Probability in Banach Spaces: Isoperimetry and Processes , 1991 .

[5]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .

[6]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[7]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[8]  R. Tibshirani.  The lasso method for variable selection in the Cox model. , 1997, Statistics in medicine.

[9]  Jiang Gui,et al.  Penalized Cox regression analysis in the high-dimensional and low-sample size settings, with applications to microarray gene expression data , 2005, Bioinform..

[10]  B. Peter.  BOOSTING FOR HIGH-DIMENSIONAL LINEAR MODELS , 2006 .

[11]  S. Geer,et al.  Classifiers of support vector machine type with \ell1 complexity regularization , 2006 .

[12]  A. Tsybakov,et al.  Sparsity oracle inequalities for the Lasso , 2007, 0705.3308.

[13]  S. Geer.  HIGH-DIMENSIONAL GENERALIZED LINEAR MODELS AND THE LASSO , 2008, 0804.0703.

[14]  F. Bunea.  Honest variable selection in linear and logistic regression models via $\ell_1$ and $\ell_1+\ell_2$ penalization , 2008, 0808.4051.

[15]  Torben Martinussen,et al.  Covariate Selection for the Semiparametric Additive Risk Model , 2009 .

[16]  Francis R. Bach,et al.  Self-concordant analysis for logistic regression , 2009, ArXiv.

[17]  P. Bickel,et al.  SIMULTANEOUS ANALYSIS OF LASSO AND DANTZIG SELECTOR , 2008, 0801.1095.

[18]  Agathe Guilloux,et al.  High-dimensional additive hazards models and the Lasso , 2011, 1106.4662.

[19]  Jianqing Fan,et al.  REGULARIZATION FOR COX'S PROPORTIONAL HAZARDS MODEL WITH NP-DIMENSIONALITY. , 2010, Annals of statistics.

[20]  Sara A. van de Geer,et al.  Classifiers of support vector machine type with \ell1 complexity regularization , 2006 .