The Dantzig selector and sparsity oracle inequalities

Let Yⱼ=f_ (Xⱼ) +ⱼ, j=1, , n, where X, X₁, , Xₙ are i. i. d. random variables in a measurable space (S, A) with distribution and, ₁, , ₙ are i. i. d. random variables with E=0 independent of (X₁, , Xₙ). Given a dictionary h₁, , h₍: S R, let f_: =₉=₁N ⱼ hⱼ, = (₁, , N) RN. Given >0, define _: =\ RN: ₁ ₊ ₍ |n^{-1₉=₁ⁿ (f_ (Xⱼ) -Yⱼ) hₖ (Xⱼ) | \} and: =^ Argmin_\|\| 䃑. In the case where f_: =f^, ^ RN, Candes and Tao Ann. Statist. 35 (2007) 2313-2351] suggested using as an estimator of ^. They called this estimator “the Dantzig selector”. We study the properties of f_ as an estimator of f_ for regression models with random design, extending some of the results of Candes and Tao (and providing alternative proofs of these results).