Local Polynomial Lp-norm Regression

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.