Asymptotic Distribution of the Improved Kernel Regression Function Estimates

Suppose that （X₁, Y₁）,…, （X_n, Y_n） is a random sample sequence from （X, Y）, If EY is finite, the regression function m （of Y on X） is defined by m（x）= E（Y|X = x）. In this paper, we obtain the asymptotic normality of the improved kernel regression function estimates \widehatm_n(x)=\sum_i=1^n Y_i I（Y_i＜b_n）K(\fracx-X_ih_n)/\sum_i=1^n Y_i IK(\fracx-X_ih_n) . Where K is a univariate density function, 0 ＜ h_n → 0 and 0 ＜b_n → ∞ （n → ∞）.