  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.
