可加模型低维分量估计平均偏差的指数界

EXPONENTIAL BOUNDS OF MEAN ERROR FOR ESTIMATES OF LOW DIMENSIONAL COMPONENT IN ADDITIVE MODEL

设 (X,Z,Y) ,(X1 ,Z1 ,Y1 ) ,… ,(Xn,Zn,Yn)为取值于 Rp× Rq× R中的i.i.d.随机向量 ,E|Y|<∞ ,Y关于 (X,Z)的回归函数 m (x,z) E(Y|(X,Z) =(x,z) )具有可加结构 :m(x,z) =m1 (x) +m2 (z) .为估计可加分量 ,采用 L inton &Nielsen(1995 )提出的直接估计法 ,给出了可加分量的最近邻估计和核估计 .在较弱的条件下 ,建立了可加分量最近邻估计和核估计的平均偏差的指数界 .

L et (X,Z,Y) ,(X1 ,Z1 ,Y1 ) ,...,(Xn,Zn,Yn) be i.i.d. random vectors taking values in Rp× Rq× R with E|Y|<∞ ,and letm(x,z) =E(Y|(X,Z) =(x,z) ) be the regression function.The additive model:m (x,z) =m1 (x ) +m2 (z) is considered.To estimate additive com ponent,the direct estimation introduced by L inton and Nielsen (1995 ) is adopted and the nearest neighbor estimate and the kernel estimate of additive component are obtained.Under mild conditions the exponential bounds of mean error for these estimates are established.