Robust estimation of error scale in nonparametric regression models

When the data used to fit a nonparametric regression model are contaminated with outliers, we need to use a robust estimator of scale in order to make robust estimation of the regression function possible. We develop a family of M  -estimators of scale constructed from consecutive differences of regression responses. Estimators in our family robustify the estimator proposed by Rice [1984. Bandwidth choice for nonparametric regression. Ann. Statist. 12, 1215–1230]. Under appropriate conditions, we establish the weak consistency and asymptotic normality of all estimators in our family. Estimators in our family vary in terms of their robustness properties. We quantify the robustness of each estimator via the maxbias. We use this measure as a basis for deriving the asymptotic breakdown point of the estimator. Our theoretical results allow us to specify conditions for estimators in our family to achieve a maximum asymptotic breakdown point of 12. We conduct a simulation study to compare the finite sample performance of our preferred M-estimator with that of three other estimators.