The Lasso for High-Dimensional Regression with a Possible Change-Point

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. Under a sparsity assumption, we derive nonasymptotic oracle inequalities for both the prediction risk and the $\ell_1$ estimation loss for regression coefficients. Furthermore, we establish conditions under which the unknown threshold parameter can be estimated at nearly $n^{-1}$ 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.