AI 中文总结
针对非参数回归中局部线性导数估计的$O(h^2)$阶偏差，本文提出迭代数据锐化方法，经$l$次锐化可将偏差阶降至$O(h^{2l+2})$，模拟实验验证其能显著降低偏差并揭示偏差-方差权衡。
AI 中文摘要
局部多项式平滑常用于非参数回归，但局部线性导数估计仍存在$O(h^2)$阶偏差。本文提出一种迭代数据锐化方法，在保留局部线性拟合简洁性的同时减少导数估计的偏差。该方法基于两个期望算子：作用于回归函数的$L_0$和作用于一阶导数的$L_1$，通过重复应用残差算子$R=I-L_0$，可构造一系列锐化后的导数估计。经过$l$次锐化步骤，偏差阶可从$O(h^2)$降至$O(h^{2l+2})$。对于高斯核，所有锐化系数均为1，得到简单的闭式单带宽表达式。对三个光滑测试函数的模拟实验表明，该方法可显著降低估计偏差，同时揭示偏差-方差权衡。
英文摘要
Local polynomial smoothing is commonly used in non-parametric regression, but local linear derivative estimation still has a bias of order $O(h^2)$. This paper proposes an iterative data sharpening method to reduce the bias of derivative estimates while retaining the simplicity of local linear fitting. The method is based on two expectation operators: $L_0$, acting on the regression function, and $L_1$, acting on the first-order derivative. By repeatedly applying the residual operator $R=I-L_0$, a series of sharpened derivative estimates can be constructed. After $l$ sharpening steps, the bias order can be reduced from $O(h^2)$ to $O(h^{2l+2})$. For the Gaussian kernel, all sharpening coefficients equal 1, giving a simple closed-form single-bandwidth expression. Simulation experiments on three smooth test functions show that this method can significantly reduce the estimation bias while revealing a bias-variance trade-off.
Comments23 pages, 4 figures, 4 tables
