This paper derives asymptotic risk (expected loss) results for shrinkage estimators with multidimensional regularization in high-dimensional settings. We introduce a class of multidimensional shrinkage estimators (MuSEs), which includes the elastic net, and show that—as the number of parameters to estimate grows—the empirical loss converges to the oracle-optimal risk. This result holds when the regularization parameters are estimated empirically via cross-validation or Stein’s unbiased risk estimate. To help guide applied researchers in their choice of estimator, we compare the empirical Bayes risk of the lasso, ridge, and elastic net in a spike and normal setting. Of the three estimators, we find that the elastic net performs best when the data are moderately sparse and the lasso performs best when the data are highly sparse. Our analysis suggests that applied researchers who are unsure about the level of sparsity in their data might benefit from using MuSEs such as the elastic net. We exploit these insights to propose a new estimator, the cubic net, and demonstrate through simulations that it outperforms the three other estimators for any sparsity level.

本文推导了在高维设置中具有多维正则化的收缩估计量的渐近风险（预期损失）结果。我们引入了一类多维收缩估计器 (MuSE)，其中包括弹性网络，并表明——随着要估计的参数数量的增加——经验损失收敛于 oracle 最优风险。当通过交叉验证或 Stein 的无偏风险估计根据经验估计正则化参数时，此结果成立。为了帮助指导应用研究人员选择估计量，我们比较了峰值和正常设置中套索、脊和弹性网的经验贝叶斯风险。在这三个估计器中，我们发现弹性网络在数据适度稀疏时表现最佳，而套索在数据高度稀疏时表现最佳。我们的分析表明，不确定数据稀疏程度的应用研究人员可能会受益于使用弹性网络等 MuSE。我们利用这些见解提出了一个新的估计量，即cubic net，并通过模拟证明它在任何稀疏度水平上都优于其他三个估计器。