Least absolute shrinkage and selection operator or Lasso is one of the widely used regularization methods in regression. Statisticians usually implement Lasso in practice by choosing the penalty parameter in a data-dependent way, the most popular being the $K-$fold cross-validation (or $K-$fold CV). However, inferential properties, such as the variable selection consistency and $n^{1/2}-$consistency, of the $K-$fold CV based Lasso estimator and validity of the Bootstrap approximation are still unknown. In this paper, we consider the heteroscedastic linear regression model and show only under some moment type conditions that the Lasso estimator with $K$-fold CV based penalty is $n^{1/2}-$consistent, but not variable selection consistent. Additionally, we establish the validity of Bootstrap in approximating the distribution of the $K-$fold CV based Lasso estimator. Therefore, our results theoretically justify the use of $K-$fold CV based Lasso estimator to perform statistical inference in linear regression. We validate our Bootstrap method for the $K-$fold CV based Lasso estimator in finite samples based on simulations. We also implement our Bootstrap based inference on a real data set.

翻译：最小绝对收缩和选择算子（Lasso）是回归分析中广泛使用的正则化方法之一。在实际应用中，统计学家通常通过数据依赖的方式选择惩罚参数来实施Lasso，其中最流行的是$K$折交叉验证（即$K$折CV）。然而，基于$K$折CV的Lasso估计量的变量选择一致性和$n^{1/2}$一致性等推断性质，以及Bootstrap近似的有效性仍属未知。本文考虑异方差线性回归模型，证明仅在矩型条件下，基于$K$折CV惩罚的Lasso估计量具有$n^{1/2}$一致性，但不具有变量选择一致性。此外，我们建立了Bootstrap近似基于$K$折CV的Lasso估计量分布的有效性。因此，我们的结果在理论上为使用基于$K$折CV的Lasso估计量进行线性回归统计推断提供了依据。我们通过模拟验证了基于$K$折CV的Lasso估计量的Bootstrap方法在有限样本中的表现，并将其推断方法应用于真实数据集。