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

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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.

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Authors and Affiliations

  1. IRMA UMR 7501, Univ. Strasbourg et CNRS, Strasbourg, France

    A. Guillou & N. Klutchnikoff

Authors
  1. A. Guillou
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  2. N. Klutchnikoff
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Correspondence to A. Guillou.

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Guillou, A., Klutchnikoff, N. Minimax pointwise estimation of an anisotropic regression function with unknown density of the design. Math. Meth. Stat. 20, 30–57 (2011). https://doi.org/10.3103/S1066530711010030

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  • DOI: https://doi.org/10.3103/S1066530711010030

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