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Simultaneous analysis of Lasso and Dantzig selector. (English) Zbl 1173.62022
Summary: We show that, under a sparsity scenario, the Lasso estimator and the Dantzig selector exhibit similar behavior. For both methods, we derive, in parallel, oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the \(\ell_p\) estimation loss for \(1\leq p\leq 2\) in the linear model when the number of variables can be much larger than the sample size.

MSC:
62G08 Nonparametric regression and quantile regression
62J05 Linear regression; mixed models
60G25 Prediction theory (aspects of stochastic processes)
62G20 Asymptotic properties of nonparametric inference
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