用Lp范数改进局部多项式回归，更适应非正态噪声。
Local Polynomial Lp-norm Regression
- 用加权Lp范数替代最小二乘法拟合局部多项式。
- 在1维和2维数据上均优于传统局部最小二乘法。
- 自动估计p值，提升对噪声分布的适应能力。
当存在非高斯噪声时，局部最小二乘回归无法达到最优效果。理论与实证均表明残差常呈现不同于正态分布的特性，因此基于其他范数的估计方法具有价值。建议使用Lp范数估计器以最小化具有非正态峰度的残差。本文提出一种局部多项式Lp范数回归方法，将局部加权最小二乘估计替换为加权Lp范数估计。同时引入一种从残差中估计参数p的新方法，增强方法的自适应性。通过数值与理论分析，证明该方法在一维数据上优于局部最小二乘法，并在二维情形下表现出良好前景。
原文摘要 · Abstract (English)
The local least squares estimator for a regression curve cannot provide optimal results when non-Gaussian noise is present. Both theoretical and empirical evidence suggests that residuals often exhibit distributional properties different from those of a normal distribution, making it worthwhile to consider estimation based on other norms. It is suggested that $L_p$-norm estimators be used to minimize the residuals when these exhibit non-normal kurtosis. In this paper, we propose a local polynomial $L_p$-norm regression that replaces weighted least squares estimation with weighted $L_p$-norm estimation for fitting the polynomial locally. We also introduce a new method for estimating the parameter $p$ from the residuals, enhancing the adaptability of the approach. Through numerical and theoretical investigation, we demonstrate our method's superiority over local least squares in one-dimensional data and show promising outcomes for higher dimensions, specifically in 2D.
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