New M-estimators in semiparametric regression with errors in variables

In the regression model with errors in variables, we observe n i.i.d. copies of (Y, Z) satisfying Y = fθ0(X) + ξ and Z = X + ε involving independent and unobserved random variables X, ξ, ε plus a regression function fθ0, known up to some finite dimensional θ. The common densities of the Xi’s and of the ξi’s are unknown whereas the distribution of ε is completely known. We aim at estimating the parameter θ by using the observations (Y1, Z1), · · · , (Yn, Zn). We propose two estimation procedures based on the least square criterion S̃θ0,g(θ) = Eθ0,g[((Y −fθ(X))w(X)] where w is some weight function, to be chosen. In the first estimation procedure, w does not depend on θ and the distribution of the ξ’s is unknown. The second estimation procedure is based on Sθ0,g(θ) = Eθ0,g[((Y − fθ(X)) − σ ξ,2)wθ(X)] where wθ is positive weight function, to be chosen, and requires the knowledge of σ ξ,2 = Var(ξ). In both cases, we propose two estimators and derive upper bounds for the risk of those estimators, depending on the smoothness of the errors density pε and on the smoothness properties of w(x)fθ(x) or wθ(x)fθ(x) with respect to x. Furthermore we give sufficient conditions that ensure that the parametric rate of convergence is achieved. We provide practical recipes for the choice of w or wθ in the case of nonlinear regression functions which are smooth on pieces allowing to gain in the order of the rate of convergence, up to the parametric rate in some cases. 1 Université Paris X, Modal’X, 200, ave. de la République, 92001 Nanterre Cedex, France 2 Université Paris VI, PMA, 175, rue de Chevaleret, 75013 Paris, France ( [email protected]) 3 Université Paris Descartes, IUT de Paris, 143 ave. de Versailles, 75016 Paris, France 4 Université Paris-Sud, Bât. 425, Département de Mathématiques, 91405 Orsay Cedex, France ([email protected]) August 23, 2008