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A sufficient condition for (C) is that the distribution of e has an even density f ( u ) that is nonincreasing for u ≥ 0 and is decreasing in a neighborhood of u = 0 (see Lemma 3.1 of Yohai (1987)). If ρ is strictly convex with a derivative ψ, then a sufficient condition is E ψ ( e ) = 0 , which for the LSE reduces to E e = 0 . 

Assumption (D) is required for ensure uniqueness of solutions. For examples (3)-(4) it is very easy to verify. For (5) it follows from the well-known linear independence of exponentials. 

If S = ∞ , (E) just means that δ < 1 (since λ$_{0}$ < ∞ by (B)). Otherwise it puts a bound on δ. In our examples we have δ = 0 , since β $^{′}$h > 0 if β has a single nonnull (positive) element. 

Assumption (F) is required in the case of non-compact A, to prevent the estimator ̂ α from “escaping to the border”. In our examples the border for the α$_{j}$ s is either zero of infinity, and (F) is easily verified by a detailed but elementary calculation (taking into account the remark above that all elements of β$_{0}$ are positive). For example, in (3) it suffices to consider neighborhoods of the form ( α$_{0}$/K, Kα$_{0}$ ) with K sufficiently large. 

Finally, (G) is easily verified for models (3)-(4)-(5). 

## 3 The results 

For separable models the M-estimate is given by 

We now state our main result. 

Theorem 1 Assume model (2) with conditions A-B-C-D-E-F-G. Then the M estimate ( ̂ α$_{n}$, ̂ β$_{n}$ ) is strongly consistent for θ$_{0}$. 

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