Bias Reduction for Local Polynomial Derivative Estimation

2026-08-13

AI Summary

This study addresses the O(h²) bias bottleneck inherent in local linear derivative estimation by proposing an iterative data sharpening method. By constructing sharpened estimators based on residual operators, this approach reduces the bias order to O(h^{2l+2}) while preserving the simplicity of local fitting. Furthermore, a closed-form single-bandwidth expression is derived specifically for Gaussian kernels. Simulation results demonstrate that the proposed algorithm significantly mitigates estimation bias and clearly elucidates the bias-variance trade-off mechanism. Consequently, this work establishes a novel paradigm for nonparametric derivative estimation that effectively balances high precision with computational efficiency, offering a robust solution to longstanding limitations in local polynomial regression techniques.

Abstract

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.

Problem

• Local polynomial smoothing
• Derivative estimation
• Bias reduction
• Non-parametric regression

Innovation

• Iterative Data Sharpening
• Bias Reduction
• Local Polynomial Derivative Estimation
• Residual Operator
• Non-parametric Regression