A NOTE ON THE CONVERGENT RATES OF M-ESTIMATES FOR A PARTLY LINEAR-MODEL

This paper concerns with M-estimators for the partly linear model Y-i = X(i)(tau) beta(o) + g(o)(T-i) + u(i), where (T-1,X(1)(tau), Y-1),... ,(T-n,X(n)(tau),Y-n) are i.i.d. random (d + 2)-vectors such that Y-i i s real-valued, X(i) epsilon R(d), and T-i ranges over a nondegenerate compact interval; u(i) is a random error; beta(o) is a d-vector of par ameters; and g(o)(.) is an unknown function. A piecewise polynomial is used to approximate g(o)(.). The estimators of beta(o) and g(o)(t) co nsidered are <(beta)over cap> and (g) over cap(n)(t) = phi(t)(tau)<(al pha)over cap> respectively, where <(alpha)over cap> and <(beta)over ca p> are the solutions of the minimization problem [GRAPHICS] and phi(.) is a vector of the basis functions of a piecewise polynomial space an d rho(.) is a function chosen suitably. Under some regular conditions, it is shown that (g) over cap(n) achieves the convergence rate which is Stone's optimal global rate of convergence of least square estimato rs for nonparametric regression and <(beta)over cap> achieves the conv ergence rate n(-1/2).