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 entries of the former estimator without penalty in the criterion function. We demonstrate via simulations that our methods are not dominated by cross-validation in terms of estimation errors and can outperform cross-validation in terms of inference. As an empirical illustration, we revisit Fryer Jr (2019), who investigated racial differences in police use of force, and confirm his findings.

翻译：本文提出了一种用于高维$\ell_{1}$惩罚M估计量惩罚参数选择的新方法，我们称之为交叉验证后自助法。我们推导了相应$\ell_1$惩罚M估计量以及后$\ell_1$惩罚M估计量的收敛速率，其中后者是在准则函数中对前一个估计量的非零项进行无惩罚的重新拟合。通过模拟实验，我们证明该方法在估计误差方面不逊于交叉验证，并在统计推断方面可能优于交叉验证。作为实证示例，我们重新审视了Fryer Jr (2019)关于警方使用武力中种族差异的研究，并证实了其结论。