Nearest Neighbor Regression with Heavy-Tailed Errors

Abstract

There has been an increasing interest in modelling regression with heavy-tailed conditional error distributions, mostly in the parametric setting. Nonparametric regression procedures have been studied almost exclusively for the cases where the conditional variance of the regressed variable is finite in the region of interest. We initiate a study of the infinite variance case. Some results in strong uniform consistency of the nearest neighbor estimator with rates are proven. The technique used provides new results and insights when higher conditional moments exist. Some asymptotic distribution theory has also been obtained when the conditional errors are in the domain of attraction of a stable law.