We consider a high dimensional regression model with a possible change point due to a covariate threshold and develop the lasso estimator of regression coefficients as well as the threshold parameter. Our lasso estimator not only selects covariates but also selects a model between linear and threshold regression models. Under a sparsity assumption, we derive non-asymptotic oracle inequalities for both the prediction risk and the l1-estimation loss for regression coefficients. Since the lasso estimator selects variables simultaneously, we show that oracle inequalities can be established without pretesting the existence of the threshold effect. Furthermore, we establish conditions under which the estimation error of the unknown threshold parameter can be bounded by a factor that is nearly n-1 even when the number of regressors can be much larger than the sample size n. We illustrate the usefulness of our proposed estimation method via Monte Carlo simulations and an application to real data.

中文翻译：

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套索用于具有可能变化点的高维回归。

我们考虑由于协变量阈值而可能具有变化点的高维回归模型，并开发回归系数的套索估计器以及阈值参数。我们的套索估计器不仅选择协变量，还选择线性和阈值回归模型之间的模型。在稀疏假设下，我们得出了回归系数的预测风险和l1估计损失的非渐近性oracle不等式。由于套索估计器同时选择变量，因此我们证明了可以建立oracle不等式，而无需预先测试阈值效应的存在。此外，我们建立了这样的条件，即即使回归变量的数量可能远大于样本大小n，未知阈值参数的估计误差也可以由接近n-1的因子限制。我们通过蒙特卡洛模拟说明了我们提出的估算方法的实用性，并将其应用于实际数据。