This paper consider penalized empirical loss minimization of convex loss functions with unknown non-linear target functions. Using the elastic net penalty we establish a finite sample oracle inequality which bounds the loss of our estimator from above with high probability. If the unknown target...

We propose a new estimator, the thresholded scaled Lasso, in high dimensional threshold regressions. First, we establish an upper bound on the sup-norm estimation error of the scaled Lasso estimator of Lee et al. (2012). This is a non-trivial task as the literature on highdimensional models has...

While variable selection and oracle inequalities for the estimation and prediction error have received considerable attention in the literature on high-dimensional models, very little work has been done in the area of testing and construction of confidence bands in high-dimensional models....