条件经验随机过程的渐近性质

设(Xi,Yi)(i=1,2,…,n)是来自总体(X,Y)的样本(独立同分布),其中X∈R1,Y∈Rq.M(x y)是Y=y时X的条件分布,Mnkn(x y)为M(x y)的第kn个最近邻域的经验分布估计量,讨论条件经验过程Sn(t,x,y)=kn12(Mnkn(x y)-M(x y))的渐近性质,得出在适当条件下,对固定的y,Sn(t,x,y)(x,t为参数)弱收敛于某一G aussian过程S(.).

An Asymptotic Property of the Conditional Empirical Process

Let(Xi,Yi)(i=1,2,…,n) be samples from the population(X,Y)(X∈R1,Y∈Rq).Given Y=y,X has the conditional distribution M(x|y).Mnkn(x|y) is the kn nearest neighbor estimator of M(x|y).The asymptotic properties of the conditional empirical process is investigated: Sn(t,x,y)=k12n(Mnkn(x|y)-M(x|y)).Then under some regular conditions we show that for fixed y,Sn(t,x,y) converges weakly to a Gaussian process S(·).