Asymptotic Normality of the Improved Kernel Estimator for Regression Function under φ -mixing Samples

Let (Xi,Yi),i≥1be a strictly stationary and φ-mixing sample sequence from (X,Y) in R^d×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)=＜b_i)h_i^-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).