Support-recovery guarantees for practical Cox–Lasso tuning rules

Establish consistent recovery of the true active set for the Cox–Lasso when the regularization parameter is selected by AIC-based tuning or cross-validation-based tuning, without relying on the oracle-type assumption of selection stability.

Background

The theoretical validity result for the model-based bootstrap after Cox–Lasso selection assumes that the selected variable set equals the true active set with probability tending to one. This oracle-type selection-stability condition permits reduction to the ordinary Cox estimator in the true active model.

The paper explicitly notes that its theory does not establish this support-recovery property for either of the practically implemented tuning strategies considered in the simulations: AIC-based tuning and the cross-validation rule selecting the penalty with minimum estimated prediction error. Proving such guarantees would connect the asymptotic bootstrap theory to the actual tuning procedures used in practice.