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

Abstract

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

Published in at http://dx.doi.org/10.3150/12-BEJ451 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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• On local $U$-statistic processes and the estimation of densities of functions of several sample variables
• Uniformly root-$N$ consistent density estimators for weakly dependent invertible linear processes