Convergence Rate of B-spline M-estimators in the Varying Coefficient Model

The model being studied in this paper is the varying coefficient model y(t)=X^T(t)β(t)+ε(t), where(y(t_ij),X_i(t_ij),t_ij) is the jth measurement of （y（t）,X（t）,t） for the ith subjects,β(t)=(β₁(t),…,β_p(t))^TT are smooth nonparametric coefficient curves. We consider B-spline M-estimators by minimizing \sum_i=1^m \sum_j=1^n_i \rho\left(y_i j-X_i^\tau\left(t_i j\right) \beta\left(t_i j\right)\right) over β（t） in a linear space of B-spline function. If the true coefficient function are smooth up to order r （r ＞ 1/2）, we show that the optimal global convergence rate of n^{-2/（2r+1）} （Stone（1985）） is attainted for the B-spline M-estimators if the number of knots is the order of n^1/（2r+1）.