The Conditional Breakdown Properties of Least Absolute Value Local Polynomial Estimators

Nonparametric regression techniques provide an effective way of identifying and examining structure in regression data. The standard approaches to nonparametric regression, such as local polynomial and smoothing spline estimators, are sensitive to unusual observations, and alternatives designed to be resistant to such observations have been proposed as a solution. Unfortunately, there has been little examination of the resistance properties of these proposed estimators. In this paper we examine the breakdown properties of local polynomial estimation based on least absolute values, rather than least squares. We show that the breakdown at any evaluation point depends on the observed distribution of observations and the kernel weight function used, and make recommendations regarding choice of kernel based on two different breakdown measures. The results suggest that the breakdown point at an evaluation point provides a useful summary of the resistance of the regression estimator to unusual observations.