Probability Theory and Related Fields

, Volume 153, Issue 1–2, pp 363–404 | Cite as

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

  • Gerard Kerkyacharian
  • Richard NicklEmail author
  • Dominique Picard


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.

Mathematics Subject Classification (2000)

62G07 60E15 42C40 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Gerard Kerkyacharian
    • 1
  • Richard Nickl
    • 2
    Email author
  • Dominique Picard
    • 1
  1. 1.Laboratoire de Probabilités et Modèles AléatoiresUniversité Paris-DiderotParisFrance
  2. 2.Statistical Laboratory, Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

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