Estimating the error distribution function in nonparametric regression with multivariate covariates

We consider nonparametric regression models with multivariate covariates and estimate the regression curve by an undersmoothed local polynomial smoother. The resulting residual-based empirical distribution function is shown to differ from the error-based empirical distribution function by the density times the average of the errors, up to a uniformly negligible remainder term. This result implies a functional central limit theorem for the residual-based empirical distribution function.