Exact constants for pointwise adaptive estimation under the Riesz transform

Jussi Klemelä; Alexandre B. Tsybakov

doi:10.1007/s00440-004-0348-9

Exact constants for pointwise adaptive estimation under the Riesz transform

Jussi Klemelä
, Alexandre B. Tsybakov

University of Heidelberg

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	441-467
Number of pages	27
Journal	Probability Theory and Related Fields
Volume	129
Issue number	3
DOIs	https://doi.org/10.1007/s00440-004-0348-9
Publication status	Published - 1 Jul 2004

Keywords

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

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10.1007/s00440-004-0348-9

Cite this

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title = "Exact constants for pointwise adaptive estimation under the Riesz transform",

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.",

keywords = "Adaptive curve estimation, Bandwidth selection, Deconvolution, Exact constants in nonparametric smoothing, Gaussian white noise, Inverse problems, Kernel estimation, Minimax risk",

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year = "2004",

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T1 - Exact constants for pointwise adaptive estimation under the Riesz transform

AU - Klemelä, Jussi

AU - Tsybakov, Alexandre B.

PY - 2004/7/1

Y1 - 2004/7/1

N2 - 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.

AB - 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.

KW - Adaptive curve estimation

KW - Bandwidth selection

KW - Deconvolution

KW - Exact constants in nonparametric smoothing

KW - Gaussian white noise

KW - Inverse problems

KW - Kernel estimation

KW - Minimax risk

U2 - 10.1007/s00440-004-0348-9

DO - 10.1007/s00440-004-0348-9

M3 - Article

AN - SCOPUS:3042813988

SN - 0178-8051

VL - 129

SP - 441

EP - 467

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3

ER -

Exact constants for pointwise adaptive estimation under the Riesz transform

Abstract

Keywords

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