Minimax pointwise estimation of an anisotropic regression function with unknown density of the design
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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.
Keywordsnonparametric regression anisotropic Hölder spaces minimax approach random design degenerate design
2000 Mathematics Subject Classification62G05 62G08
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- 1.N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation (Cambridge Univ. Press, 1987).Google Scholar
- 2.J. Fan and I. Gijbels, Local Polynomial Modelling and Its Applications, in Monographs on Statist. and Appl. Probab. (Chapman and Hall, London, 1996).Google Scholar
- 5.S. Gaïffas and G. Lecué, “Aggregation of Penalized Empirical Risk Minimizers in Regression”, (2009) (in press).Google Scholar
- 10.N. Klutchnikoff, Sur l’estimation adaptative de fonctions anisotropes, PhD thesis (Marseille, 2005).Google Scholar
- 16.A. B. Tsybakov, Introduction à l’estimation non-paramétrique, in Mathématiques & Applications (Springer, Berlin, 2004), Vol. 41.Google Scholar