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Mathematical Methods of Statistics

, Volume 20, Issue 1, pp 30–57 | Cite as

Minimax pointwise estimation of an anisotropic regression function with unknown density of the design

  • A. GuillouEmail author
  • N. Klutchnikoff
Article

Abstract

Our aim in this paper is to estimate with best possible accuracy an unknown multidimensional regression function at a given point where the design density is also unknown. To reach this goal, we will follow the minimax approach: it will be assumed that the regression function belongs to a known anisotropic Hölder space. In contrast to the parameters defining the Hölder space, the density of the observations is assumed to be unknown and will be treated as a nuisance parameter. New minimax rates are exhibited as well as local polynomial estimators which achieve these rates. As these estimators depend on a tuning parameter, the problem of its selection is also discussed.

Keywords

nonparametric regression anisotropic Hölder spaces minimax approach random design degenerate design 

2000 Mathematics Subject Classification

62G05 62G08 

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Copyright information

© Allerton Press, Inc. 2011

Authors and Affiliations

  1. 1.IRMA UMR 7501Univ. Strasbourg et CNRSStrasbourgFrance

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