Local Polynomial Lp-norm Regression

• URL: http://arxiv.org/abs/2504.18695v1
• Date: Fri, 25 Apr 2025 21:04:19 GMT
• Title: Local Polynomial Lp-norm Regression
• Authors: Ladan Tazik, James Stafford, John Braun,
• Abstract summary: Local least squares estimation cannot provide optimal results when non-Gaussian noise is present.<n>It is suggested that $L_p$-norm estimators be used to minimize the residuals when these exhibit non-normal kurtosis.<n>We show our method's superiority over local least squares in one-dimensional data and show promising outcomes for higher dimensions, specifically in 2D.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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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