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

We consider a nonparametric regression model Y = r(X) + ε with a random covariate X that is independent of the error ε. Then the density of the response Y is a convolution of the densities of ε 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 C₀(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.