Anisotropic adaptive kernel deconvolution

F. Comte; C. Lacour

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 2, page 569-609
  • ISSN: 0246-0203

Abstract

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In this paper, we consider a multidimensional convolution model for which we provide adaptive anisotropic kernel estimators of a signal density f measured with additive error. For this, we generalize Fan’s (Ann. Statist.19(3) (1991) 1257–1272) estimators to multidimensional setting and use a bandwidth selection device in the spirit of Goldenshluger and Lepski’s (Ann. Statist.39(3) (2011) 1608–1632) proposal for density estimation without noise. We consider first the pointwise setting and then, we study the integrated risk. Our estimators depend on an automatically selected random bandwidth. We assume both ordinary and super smooth components for measurement errors, which have known density. We also consider both anisotropic Hölder and Sobolev classes for f . We provide nonasymptotic risk bounds and asymptotic rates for the resulting data driven estimator, together with lower bounds in most cases. We provide an illustrative simulation study, involving the use of Fast Fourier Transform algorithms. We conclude by a proposal of extension of the method to the case of unknown noise density, when a preliminary pure noise sample is available.

How to cite

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Comte, F., and Lacour, C.. "Anisotropic adaptive kernel deconvolution." Annales de l'I.H.P. Probabilités et statistiques 49.2 (2013): 569-609. <http://eudml.org/doc/271987>.

@article{Comte2013,
abstract = {In this paper, we consider a multidimensional convolution model for which we provide adaptive anisotropic kernel estimators of a signal density $f$ measured with additive error. For this, we generalize Fan’s (Ann. Statist.19(3) (1991) 1257–1272) estimators to multidimensional setting and use a bandwidth selection device in the spirit of Goldenshluger and Lepski’s (Ann. Statist.39(3) (2011) 1608–1632) proposal for density estimation without noise. We consider first the pointwise setting and then, we study the integrated risk. Our estimators depend on an automatically selected random bandwidth. We assume both ordinary and super smooth components for measurement errors, which have known density. We also consider both anisotropic Hölder and Sobolev classes for $f$. We provide nonasymptotic risk bounds and asymptotic rates for the resulting data driven estimator, together with lower bounds in most cases. We provide an illustrative simulation study, involving the use of Fast Fourier Transform algorithms. We conclude by a proposal of extension of the method to the case of unknown noise density, when a preliminary pure noise sample is available.},
author = {Comte, F., Lacour, C.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {adaptive kernel estimator; anisotropic estimation; deconvolution; density estimation; measurement errors; multidimensional},
language = {eng},
number = {2},
pages = {569-609},
publisher = {Gauthier-Villars},
title = {Anisotropic adaptive kernel deconvolution},
url = {http://eudml.org/doc/271987},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Comte, F.
AU - Lacour, C.
TI - Anisotropic adaptive kernel deconvolution
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 2
SP - 569
EP - 609
AB - In this paper, we consider a multidimensional convolution model for which we provide adaptive anisotropic kernel estimators of a signal density $f$ measured with additive error. For this, we generalize Fan’s (Ann. Statist.19(3) (1991) 1257–1272) estimators to multidimensional setting and use a bandwidth selection device in the spirit of Goldenshluger and Lepski’s (Ann. Statist.39(3) (2011) 1608–1632) proposal for density estimation without noise. We consider first the pointwise setting and then, we study the integrated risk. Our estimators depend on an automatically selected random bandwidth. We assume both ordinary and super smooth components for measurement errors, which have known density. We also consider both anisotropic Hölder and Sobolev classes for $f$. We provide nonasymptotic risk bounds and asymptotic rates for the resulting data driven estimator, together with lower bounds in most cases. We provide an illustrative simulation study, involving the use of Fast Fourier Transform algorithms. We conclude by a proposal of extension of the method to the case of unknown noise density, when a preliminary pure noise sample is available.
LA - eng
KW - adaptive kernel estimator; anisotropic estimation; deconvolution; density estimation; measurement errors; multidimensional
UR - http://eudml.org/doc/271987
ER -

References

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