标题： $K$重交叉验证基于惩罚的方法是否执行变量选择，还是导致 Lasso 的$n^{1/2}$一致性？ 显示英文标题

标题： Does $K$-fold CV based penalty perform variable selection or does it lead to $n^{1/2}$-consistency in Lasso?

摘要： 最小绝对收缩和选择算子或Lasso，由Tibshirani（1996）引入，是回归中广泛使用的正则化方法之一。观察到Lasso的性质会随着惩罚参数的选择而剧烈变化。Lahiri（2021）的最新结果表明，根据惩罚参数的性质，Lasso可以是变量选择一致的，也可以是$n^{1/2}-$一致的。然而，实践者通常以数据依赖的方式选择惩罚参数，最流行的是$K$折交叉验证。在本文中，我们探讨当惩罚参数基于$K$折交叉验证选择且$K$固定时，Lasso的变量选择一致性和$n^{1/2}-$一致性。我们考虑固定维数的异方差线性回归模型，并证明基于$K$折交叉验证的惩罚的Lasso是$n^{1/2}-$一致的，但不是变量选择一致的。 我们还建立了基于$K$折交叉验证的惩罚的$n^{1/2}-$一致性作为中间结果。 此外，作为$n^{1/2}-$一致性的结果，我们建立了 Bootstrap 方法的有效性，以近似基于$K-$折交叉验证的 Lasso 估计量的分布。 我们根据一个中等规模的模拟研究验证了有限样本中的 Bootstrap 近似效果。 因此，我们的结果基本上证明了在实践中使用$K$折交叉验证来基于 Lasso 回归中的$n^{1/2}-$缩放枢轴量进行推断的合理性。 显示英文摘要

摘要： Least absolute shrinkage and selection operator or Lasso, introduced by Tibshirani (1996), is one of the widely used regularization methods in regression. It is observed that the properties of Lasso vary wildly depending on the choice of the penalty parameter. The recent results of Lahiri (2021) suggest that, depending on the nature of the penalty parameter, Lasso can either be variable selection consistent or be $n^{1/2}-$consistent. However, practitioners generally implement Lasso by choosing the penalty parameter in a data-dependent way, the most popular being the $K$-fold cross-validation. In this paper, we explore the variable selection consistency and $n^{1/2}-$consistency of Lasso when the penalty is chosen based on $K$-fold cross-validation with $K$ being fixed. We consider the fixed-dimensional heteroscedastic linear regression model and show that Lasso with $K$-fold cross-validation based penalty is $n^{1/2}-$consistent, but not variable selection consistent. We also establish the $n^{1/2}-$consistency of the $K$-fold cross-validation based penalty as an intermediate result. Additionally, as a consequence of $n^{1/2}-$consistency, we establish the validity of Bootstrap to approximate the distribution of the Lasso estimator based on $K-$fold cross-validation. We validate the Bootstrap approximation in finite samples based on a moderate simulation study. Thus, our results essentially justify the use of $K$-fold cross-validation in practice to draw inferences based on $n^{1/2}-$scaled pivotal quantities in Lasso regression.