非参数回归估计中CV和GCV准则的渐近最优性

Asymptotic Optimality of CV and GCV Criterions in Nonparametric Regression Estimation

设非参数回归模型y_i=f(x_i)+ε_i,i=1,…,n,f(x)是[0,1]上的未知的非参数回归函数,f(x)的核估计具有一个光滑参数h,分别利用CV和GCV准则来选择参数h,得到f(x)的核估计及相应的Stein估计,本文证明了这类估计在强收敛意义下是渐近最优的。

Suppose nonparametric regression model to be Yi = f(xi) + εi, i = 1,... , n. f (x) is an unknown nonparametric regression function on [0,1], the kernal estimator of f(x) has a smoothing parameter h, use CV and GCV criterions to choose the parameter h respectively, the kernal estimators and the associated Stein's estimators are obtained, the paper proves these estimators are asymtotically optimal with respect to strong convergency.