Oracle inequalities for the lasso in the Cox model

TL;DR

This paper establishes sharper oracle inequalities for the Lasso estimator in high-dimensional Cox models with time-dependent covariates, demonstrating that key matrix factors are bounded away from zero under mild conditions.

Contribution

It introduces improved oracle inequalities for the Lasso in Cox models by leveraging compatibility and cone invertibility factors, which are shown to be positive constants in high-dimensional, time-dependent settings.

Findings

Oracle inequalities are sharper than previous results.

Compatibility and cone invertibility factors are positive constants.

Results hold even when covariate number exceeds sample size.

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.