变系数模型B样条M估计的收敛性

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

考虑变系数模型y(t)=XT(t)β(t)+ε(t).设(y(tij),Xi(tij),tij)是第i个个体的第j次观察.函数系数β(t)=(β1(t),…,βp(t))T是光滑的非参数函数向量,在B样条的函数空间上最小化得到β(t)的B样条M估计.若βk(t),k=1,…,p是r(r>1/2)阶光滑的,证得若结点的数目是O(n1/(2r+1)),则β(t)的B样条M估计达到最优的收敛速度O(n-r/(2r+1))(Stone(1985)).

The model being studied in this paper is the varying coefficient model y(t) = XT(t)β(t) + ε(t), where (y(tij),Xi(tij),tij) is the jth measurement of (y(t),X(t),t) for the ith subjects, β(t) = (β1 (t),… ,βP(t))T are smooth nonparametric coefficient curves. We consider B-spline M-estimators by minimizing XiT(tij)β(tij)) 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 n1/(2r+1).