In this paper we propose a variable bandwidth kernel regression estimator for $i.i.d.$ observations in $\mathbb{R}^2$ to improve the classical Nadaraya-Watson estimator. The bias is improved to the order of $O(h_n^4)$ under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kernel method over the classical kernel method.

翻译：在本文中,我们建议用$1.i.d.d.d.d.d.d.d.subs 来设置一个可变带宽内核回归估计值,用于改善古典Nadaraya-Watson 估计值。这种偏差被改进为$O(h_n ⁇ 4),条件是密度函数的第五顺序衍生物和回归函数的第六顺序衍生物受约束和连续。我们还为拟议的理想和真实的可变内核回归估计值设定了核心限值。模拟研究证实了我们的结果,并展示了可变带内核法相对于经典内核法的优势。