A Note on the Convergent Rates of M-Estimates for a Partly Linear Model

Abstract

This paper concerns with M-estimators for the partly linear model , where are i.i.d. random (d + 2)-vectors such that Y i , is real-valued, X i ∊ R d , and T i ranges over a nondegenerate compact interval; u i , is a random error; β o is a d-vector of parameters; and g o (·) is an unknown function. A piecewise polynomial is used to approximate g o (·). The estimators of β o and g o (t) considered are and respectively, where and are the solutions of the minimization problem and ϕ(·) is a vector of the basis functions of a piecewise polynomial space and ρ(·) is a function chosen suitably. Under some regular conditions, it is shown that achieves the convergence rate which is Stone’s optimal global rate of convergence of least square estimators for nonparametric regression and achieves the convergence rate n -1/2.