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Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds
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  • Published: 18 March 2011

Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds

  • Gerard Kerkyacharian1,
  • Richard Nickl2 &
  • Dominique Picard1 

Probability Theory and Related Fields volume 153, pages 363–404 (2012)Cite this article

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Abstract

Let X 1, . . . , X n be a random sample from some unknown probability density f defined on a compact homogeneous manifold M of dimension d ≥ 1. Consider a ‘needlet frame’ \({\{\phi_{j\eta}\}}\) describing a localised projection onto the space of eigenfunctions of the Laplace operator on M with corresponding eigenvalues less than 22j, as constructed in Geller and Pesenson (J Geom Anal 2011). We prove non-asymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator f n (j) obtained from an empirical estimate of the needlet projection \({\sum_\eta \phi_{j \eta}\int f \phi_{j \eta}}\) of f. We apply these results to construct risk-adaptive estimators and nonasymptotic confidence bands for the unknown density f. The confidence bands are adaptive over classes of differentiable and Hölder-continuous functions on M that attain their Hölder exponents.

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Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Paris-Diderot, 175, Rue de Chevaleret, 75013, Paris, France

    Gerard Kerkyacharian & Dominique Picard

  2. Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB30WB, UK

    Richard Nickl

Authors
  1. Gerard Kerkyacharian
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  2. Richard Nickl
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  3. Dominique Picard
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Correspondence to Richard Nickl.

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Kerkyacharian, G., Nickl, R. & Picard, D. Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds. Probab. Theory Relat. Fields 153, 363–404 (2012). https://doi.org/10.1007/s00440-011-0348-5

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  • Received: 08 November 2010

  • Revised: 11 February 2011

  • Published: 18 March 2011

  • Issue Date: June 2012

  • DOI: https://doi.org/10.1007/s00440-011-0348-5

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Mathematics Subject Classification (2000)

  • 62G07
  • 60E15
  • 42C40
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