Local Polynomial Lp-norm Regression
arXiv:2504.18695v1 Announce Type: new
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.
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.
Ladan Tazik (Dept. of Computer Science, Mathematics, Physics and Statistics, University of British Columbia, Okanagan campus), James Stafford (Dept. of Statistical Sciences, University of Toronto), John Braun (Dept. of Computer Science, Mathematics, Physics and Statistics, University of British Columbia, Okanagan campus)
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