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Nonparametric regression estimation based on spatiallyinhomogeneous data: minimax global convergence rates and adaptivity

Published online by Cambridge University Press:  06 December 2013

Anestis Antoniadis ,
Marianna Pensky  and
Theofanis Sapatinas
Anestis Antoniadis
Affiliation:
Laboratoire Jean Kuntzmann, Universite Joseph Fourier, 38041 Grenoble Cedex 9, France. [email protected]
Marianna Pensky
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, 32816-1364, USA; [email protected]
Theofanis Sapatinas
Affiliation:
Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, CY 1678 Nicosia, Cyprus; [email protected]
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Abstract

We consider the nonparametric regression estimation problem of recovering an unknownresponse function f on the basis of spatially inhomogeneous data when thedesign points follow a known density g with a finite number ofwell-separated zeros. In particular, we consider two different cases: wheng has zeros of a polynomial order and when g has zerosof an exponential order. These two cases correspond to moderate and severe data losses,respectively. We obtain asymptotic (as the sample size increases) minimax lower bounds forthe L2-risk when f is assumed to belong to aBesov ball, and construct adaptive wavelet thresholding estimators of fthat are asymptotically optimal (in the minimax sense) or near-optimal within alogarithmic factor (in the case of a zero of a polynomial order), over a wide range ofBesov balls. The spatially inhomogeneous ill-posed problem that we investigate isinherently more difficult than spatially homogeneous ill-posed problems like,e.g., deconvolution. In particular, due to spatial irregularity,assessment of asymptotic minimax global convergence rates is a much harder task than thederivation of asymptotic minimax local convergence rates studied recently in theliterature. Furthermore, the resulting estimators exhibit very different behavior andasymptotic minimax global convergence rates in comparison with the solution of spatiallyhomogeneous ill-posed problems. For example, unlike in the deconvolution problem, theasymptotic minimax global convergence rates are greatly influenced not only by the extentof data loss but also by the degree of spatial homogeneity of f.Specifically, even if 1/g is non-integrable, one can recoverf as well as in the case of an equispaced design (in terms ofasymptotic minimax global convergence rates) when it is homogeneous enough since theestimator is “borrowing strength” in the areas where f is adequatelysampled.

Keywords

AdaptivityBesov spacesinhomogeneous dataminimax estimationnonparametric regressionthresholdingwavelet estimation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 1 - 41
Copyright
© EDP Sciences, SMAI 2013

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