  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.
