Resistant dispersion estimation for nonparametric regression

I. Kalogridis

{}^{1}

and S. Van Aelst

{}^{2}

Abstract

Estimation of the dispersion of the errors is a central problem in regression analysis. An estimate of this dispersion is needed for most statistical inference procedures such as the construction of confidence intervals. In the context of robustness it also plays a crucial role in the identification of outliers. Several nonparametric methods to estimate the dispersion function in heteroscedastic regression models have been proposed through the years. However, the vast majority of them rely on Gaussian likelihood and least-squares procedures, leading to estimators that are sensitive to atypical observations such as gross errors in the response space. To remedy this deficiency, a novel class of resistant nonparametric dispersion estimators is introduced and studied. This class of estimators builds upon the likelihood principle and spline smoothing. Estimators in this class can combine resistance towards atypical observations with high efficiency at the Gaussian model. It is shown that the new class of estimators is computationally efficient and enjoys optimal asymptotic properties. Its highly competitive performance is illustrated through a simulation study and a real-data example containing atypical observations

${}^{1}$

School of Mathematics and Statistics, University of Glasgow, UK [ioannis.kalogridis@glasgow.ac.uk]
${}^{2}$

Department of Mathematics, KU Leuven, Belgium [stefan.vanaelst@kuleuven.be]

Keywords: Robustness, dispersion, nonparametric regression, asymptotics