Uniform convergence of convolution estimators for the response density in nonparametric regression

Abstract

We consider a nonparametric regression model \(Y=r(X)+\varepsilon\) with a random covariate \(X\) that is independent of the error \(\varepsilon\). Then the density of the response \(Y\) is a convolution of the densities of \(\varepsilon\) and \(r(X)\). It can therefore be estimated by a convolution of kernel estimators for these two densities, or more generally by a local von Mises statistic. If the regression function has a nowhere vanishing derivative, then the convolution estimator converges at a parametric rate. We show that the convergence holds uniformly, and that the corresponding process obeys a functional central limit theorem in the space \(C_0(\mathbb {R})\) of continuous functions vanishing at infinity, endowed with the sup-norm. The estimator is not efficient. We construct an additive correction that makes it efficient.