Oracle Inequalities and Selection Consistency for Weighted Lasso in High-dimensional Additive Hazards Model

The additive hazards model has many applications in high-throughput genomic data analysis and clinical studies. In this article, we study the weighted Lasso estimator for the additive hazards model in sparse, high-dimensional settings where the number of time-dependent covariates is much larger than the sample size. Based on compatibility, cone invertibility factors, and restricted eigenvalues of the Hessian matrix, we establish some non-asymptotic oracle inequalities for the weighted Lasso. Under mild conditions, we show that these quantities are bounded from below by positive constants, thus the compatibility and cone invertibility factors can be treated as positive constants in the oracle inequalities. A multistage adaptive method with weights recursively generated from a concave penalty is presented. We prove a selection consistency theorem and establish an upper bound for dimension of the weighted Lasso estimator.