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Exact constants for pointwise adaptive estimation under the Riesz transform
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  • Published: 29 April 2004

Exact constants for pointwise adaptive estimation under the Riesz transform

  • Jussi Klemelä1 &
  • Alexandre B. Tsybakov2 

Probability Theory and Related Fields volume 129, pages 441–467 (2004)Cite this article

Abstract.

We consider nonparametric estimation of a multivariate function and its partial derivatives at a fixed point when the Riesz transform of the function is observed in Gaussian white noise. We assume that the unknown function belongs to some Sobolev class and construct an estimation procedure which achieves the best asymptotic minimax risk when the smoothness of the function is unknown.

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Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, 69120, Heidelberg, Germany

    Jussi Klemelä

  2. Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 6, 4 pl.Jussieu, 188, 75252, Paris, France

    Alexandre B. Tsybakov

Authors
  1. Jussi Klemelä
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  2. Alexandre B. Tsybakov
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Corresponding author

Correspondence to Jussi Klemelä.

Additional information

Writing of this article was financed by Deutsche Forschungsgemeinschaft under project MA1026/6-2 and Rolf Nevanlinna Institute.

Mathematics Subject Classification (2000): 62G05, 62G20

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Klemelä, J., Tsybakov, A. Exact constants for pointwise adaptive estimation under the Riesz transform. Probab. Theory Relat. Fields 129, 441–467 (2004). https://doi.org/10.1007/s00440-004-0348-9

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  • Received: 15 November 2001

  • Revised: 18 September 2003

  • Published: 29 April 2004

  • Issue Date: July 2004

  • DOI: https://doi.org/10.1007/s00440-004-0348-9

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Key words or phrases:

  • Adaptive curve estimation
  • Bandwidth selection
  • Deconvolution
  • Exact constants in nonparametric smoothing
  • Gaussian white noise
  • Inverse problems
  • Kernel estimation
  • Minimax risk
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