Let
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Yj=f*(Xj)+ξj, j=1, …, n,
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where X, X1, …, Xn are i.i.d. random variables in a measurable space $(S,\mathcal{A})$ with distribution Π and ξ, ξ1, …, ξn are i.i.d. random variables with ${\mathbb{E}}\xi=0$ independent of (X1, …, Xn). Given a dictionary h1, …, hN: S↦ℝ, let fλ:=∑j=1Nλjhj, λ=(λ1, …, λN)∈ℝN. Given ɛ>0, define
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̂Λɛ:={λ∈ℝN: max1≤k≤N|n−1∑j=1n(fλ(Xj)−Yj)hk(Xj)|≤ɛ}
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and
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̂λ:=̂λɛ∈Argminλ∈̂Λɛ‖λ‖ℓ1.
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In the case where f*:=fλ*, λ*∈ℝN, Candes and Tao [Ann. Statist. 35 (2007) 2313–2351] suggested using ̂λ as an estimator of λ*. They called this estimator “the Dantzig selector”. We study the properties of f̂λ 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).