Simultaneous Lasso and Dantzig Selector in High Dimensional Nonparametric Regression

During the last few years, a great deal of attention has been focused on Lasso and Dantzig selector in high dimensional linear regression when the number of variables can be much larger than the sample size. Under a sparsity scenario, Bickel et al. (2009) showed that the Lasso estimator and the Dantzig selector exhibit similar behavior, and derived oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the L-p estimation loss in the linear model. The Assumption RE(s, c) and Assumption RE(s,m,c) play a significant role in their paper. In this paper, the assumptions equivalent with Assumption RE(s, c) and Assumption RE (s,m,c) are given. More precise oracle inequalities for the prediction risk in the general nonparametric regression model and bounds on the L-p estimation loss in the linear model are derived when the number of variables can be much larger than the sample size.