We develop a new method for selecting the penalty parameter for $\ell_1$-penalized M-estimators in high dimensions, which we refer to as bootstrapping after cross-validation. We derive rates of convergence for the corresponding $\ell_1$-penalized M-estimator and also for the post-$\ell_1$-penalized M-estimator, which refits the non-zero parameters of the former estimator without penalty in the criterion function. We demonstrate via simulations that our method is not dominated by cross-validation in terms of estimation errors and outperforms cross-validation in terms of inference. As an illustration, we revisit Fryer Jr (2019), who investigated racial differences in police use of force, and confirm his findings.

翻译：我们提出了一种新方法，用于在高维场景下选择ℓ1惩罚M估计器的惩罚参数，该方法被称为“交叉验证后自助法”。我们推导了相应ℓ1惩罚M估计器以及后ℓ1惩罚M估计器的收敛速率，后者在无惩罚条件下对前者估计器的非零参数进行重新拟合。通过仿真实验，我们证明该方法在估计误差方面不劣于交叉验证法，并在统计推断方面优于交叉验证法。作为应用实例，我们重新审视了Fryer Jr（2019）关于警察执法中种族差异的研究，并证实了其结论。