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Schick, Anton and Wefelmeyer, Wolfgang (2013). Uniform convergence of convolution estimators for the response density in nonparametric regression. Bernoulli, 19 (5B). S. 2250 - 2277. VOORBURG: INT STATISTICAL INST. ISSN 1573-9759

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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(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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Schick, Anton UNSPECIFIED UNSPECIFIED UNSPECIFIED Wefelmeyer, Wolfgang UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-472283
DOI:	10.3150/12-BEJ451
Journal or Publication Title:	Bernoulli
Volume:	19
Number:	5B
Page Range:	S. 2250 - 2277
Date:	2013
Publisher:	INT STATISTICAL INST
Place of Publication:	VOORBURG
ISSN:	1573-9759
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language ROOT-N CONSISTENT; MOVING AVERAGE PROCESSES; NONLINEAR-REGRESSION; EFFICIENT ESTIMATION; MARGINAL DENSITY; DERIVATIVES; VARIABLES; MODELS Multiple languages Statistics & Probability Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/47228