Let h ( x , α ) = ( h$_{1}$ ( x , α ) , ..., h$_{p}$$_{2}$ ( x , α )) ′ where in general a ′ denotes the transpose of a . The necessary assumptions are: A B is a closed set such that tβ ∈ B for all β ∈ B and t > 0 . B sup$_{α}$$_{∈}$$_{A}$ E | ρ ( y − β $^{′}$h ( x , α )) | < ∞ for all β ∈ B. C The function E ρ ( e − t ) –where e denotes any copy of e$_{i}$ – has a unique min- imum at t = 0 . Put λ$_{0}$ = E ρ ( e ) . D h is continuous in α a.s. and E Let S = sup$_{t}$ ρ ( t ) (which may be infinite). Then α = α$_{0}$ ⇒ sup β ∈ B P { β $^{′}$h ( x , α ) = β ′ $_{0}$h ( x , α$_{0}$ ) } < 1 (9) δ =: sup β = 0 , α ∈ A P ( β $^{′}$h ( x , α ) = 0) < 1 − λ$_{0}$ S . (10) sup β inf U ∈U sup α/ ∈ U P { β $^{′}$h ( x , α ) = β ′ $_{0}$h ( x , α$_{0}$ ) } < 1 . F Call U the family of all open neighborhoods of α$_{0}$. Then G h is bounded as a function of α, i.e., sup$_{α}$$_{∈}$$_{A}$ ‖ h ( x , α ) ‖ < ∞ a . s . We now comment on the assumptions. For (A) to hold in examples (3)-(4)-(5) we must enlarge the range of β$_{j}$ s to β$_{j}$ ≥ 0 . However, to ensure the validity of (D) and (F), it will be assumed that the elements of the “true” vector β$_{0}$ are all positive. If ρ is bounded, (B) holds without further conditions. Sufficient conditions for Huber’s ρ and for the LSE are finite moments of e and of h ( x , α ) , of orders one and two, respectively. 5