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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
Article

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.

Mathematics Subject Classification (2000)

62G07 60E15 42C40 

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References

  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special functions. In: Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  2. 2.
    Baldi P., Kerkyacharian G., Marinucci D., Picard D.: Adaptive density estimation for directional data using needlets. Ann. Stat. 37(6A), 3362–3395 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bickel P.J., Rosenblatt M.: On some global measures of the deviations of density function estimates. Ann. Stat. 1, 1071–1095 (1973) ISSN:0090-5364MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Dümbgen L.: Optimal confidence bands for shape-restricted curves. Bernoulli 9(3), 423–449 (2003) ISSN:1350-7265MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Faraut, J.: Analysis on Lie groups. In: Cambridge Studies in Advanced Mathematics, vol. 110. Cambridge University Press, Cambridge (2008). ISBN:978-0-521-71930-8Google Scholar
  6. 6.
    Frisch, U., Parisi, G.: On the singularity structure of fully developed turbulence; appendix to fully developed turbulence and intermittency. In: Proc. Int. Summer School Phys. Enrico Fermi (1985)Google Scholar
  7. 7.
    Geller D., Mayeli A.: Continuous wavelets on compact manifolds. Math. Z. 262(4), 895–927 (2009) ISSN:0025-5874MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Geller D., Pesenson I.Z.: Band-limited localized parseval frames and besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21, 334–371 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Giné E.: The addition formula for the eigenfunctions of the Laplacian. Adv. Math. 18(1), 102–107 (1975) ISSN:0001-8708zbMATHCrossRefGoogle Scholar
  10. 10.
    Giné E., Nickl R.: Uniform limit theorems for wavelet density estimators. Ann. Probab. 37, 1605–1646 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Giné E., Nickl R.: Confidence bands in density estimation. Ann. Stat. 38, 1122–1170 (2010)zbMATHCrossRefGoogle Scholar
  12. 12.
    Giné E., Nickl R.: Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections. Bernoulli 16, 1137–1163 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. In: Pure and Applied Mathematics, vol. 80 (1978)Google Scholar
  14. 14.
    Helgason, S.: Groups and geometric analysis. In: Mathematical Surveys and Monographs, vol. 83. American Mathematical Society, Providence (2000). ISBN:0-8218-2673-5 [Integral geometry, invariant differential operators, and spherical functions, Corrected reprint of the 1984 original]Google Scholar
  15. 15.
    Jaffard S.: On the Frisch-Parisi conjecture. J. Math. Pures Appl. (9) 79(6), 525–552 (2000) ISSN:0021-7824MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kerkyacharian, G., Pham Ngoc, T.M., Picard, D.: Localized deconvolution on the sphere. Ann. Stat. (2011, to appear)Google Scholar
  17. 17.
    Klemelä J.: Asymptotic minimax risk for the white noise model on the sphere. Scand. J. Stat. 26(3), 465–473 (1999) ISSN:0303-6898zbMATHCrossRefGoogle Scholar
  18. 18.
    Koltchinskii V.: Local Rademacher complexities and oracle inequalities in risk minimization. Ann. Stat. 34(6), 2593–2656 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Leeb H., Pötscher B.M.: Can one estimate the conditional distribution of post-model-selection estimators?. Ann. Stat. 34(5), 2554–2591 (2006)zbMATHCrossRefGoogle Scholar
  20. 20.
    Lepskiĭ O.V.: Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Teor. Veroyatnost. i Primenen. 36(4), 645–659 (1991) ISSN:0040-361XMathSciNetGoogle Scholar
  21. 21.
    Lounici K., Nickl R.: Global uniform risk bounds for wavelet deconvolution estimators. Ann. Stat. 39, 201–231 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Low M.: On nonparametric confidence intervals. Ann. Stat. 25(6), 2547–2554 (1997) ISSN:0090-5364MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Massart, P.: Concentration inequalities and model selection. Lecture Notes in Mathematics, vol. 1896. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003, With a foreword by Jean Picard. Springer (2007). ISBN:978-3-540-48497-4; 3-540-48497-3Google Scholar
  24. 24.
    Narcowich F., Petrushev P., Ward J.: Local tight frames on spheres. SIAM J. Math. Anal. 38, 574–594 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Narcowich F.J., Petrushev P., Ward J.: Decomposition of Besov and Triebel-Lizorkin spaces on the sphere. J. Funct. Anal. 238, 530–564 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Picard D., Tribouley K.: Adaptive confidence interval for pointwise curve estimation. Ann. Stat. 28(1), 298–335 (2000) ISSN:0090-5364MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Stein E., Weiss G.: Introduction to Fourier Analysis on Euclidean spaces. Princeton University Press, Princeton (1971)zbMATHGoogle Scholar
  28. 28.
    Talagrand M.: New concentration inequalities in product spaces. Invent. Math. 126(3), 505–563 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Wang H.-C.: Two-point homogeneous spaces. Ann. Math. (2) 55, 177–191 (1952) ISSN:0003-486XzbMATHCrossRefGoogle Scholar
  30. 30.
    Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups, vol. 94. Springer-Verlag, New York (1983). ISBN:0-387-90894-3. Corrected reprint of the 1971 editionGoogle Scholar

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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