## Bernoulli

• Bernoulli
• Volume 19, Number 5B (2013), 2250-2276.

### 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.

#### Article information

Source
Bernoulli, Volume 19, Number 5B (2013), 2250-2276.

Dates
First available in Project Euclid: 3 December 2013

https://projecteuclid.org/euclid.bj/1386078602

Digital Object Identifier
doi:10.3150/12-BEJ451

Mathematical Reviews number (MathSciNet)
MR3160553

Zentralblatt MATH identifier
1281.62103

#### Citation

Schick, Anton; Wefelmeyer, Wolfgang. Uniform convergence of convolution estimators for the response density in nonparametric regression. Bernoulli 19 (2013), no. 5B, 2250--2276. doi:10.3150/12-BEJ451. https://projecteuclid.org/euclid.bj/1386078602

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