φ-混合下回归函数改良核估计的渐近正态性
Asymptotic Normality of the Improved Kernel Estimator for Regression Function under φ -mixing Samples
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摘要: 设(Xi,Yi),i≥1是从取值于Rd×R的总体(X,Y)中抽取的严平稳、φ-混合样本.回归函数m(x)=E(Y|X=x)改良的递归核估计定义为\widehatm_n^(2)(x)=\left\sum_i=1^n Y_i I\left(\left|Y_i\right|\right.\right.\widehatm_n^(2)(x)=＜bi)hi-d\left.h_i^-d K\left(\fracx-X_ih_i\right)\right /\left\sum_j=1^n h_j^-d k\left(\fracx-X_jh_j\right)\right.本文在适当的条件下,讨论了\widehatm_n^(2)(x)的渐近正态性.Abstract: Let (Xi,Yi),i≥1be a strictly stationary and φ-mixing sample sequence from (X,Y) in Rd×R. The improved recursive kernel estimator of regression function m(x)=E(Y|X=x) is defined by \widehatm_n^(2)(x)=\left\sum_i=1^n Y_i I\left(\left|Y_i\right|\right.\right.\widehatm_n^(2)(x)=＜bi)hi-d\left.h_i^-d K\left(\fracx-X_ih_i\right)\right /\left\sum_j=1^n h_j^-d k\left(\fracx-X_jh_j\right)\right. Under suitable conditions, we prove the asymptotic normality of \widehatm_n^(2)(x).
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