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Block Thresholding on the Sphere

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Abstract

The aim of this paper is to study nonparametric regression estimators on the sphere based on needlet block thresholding. The block thresholding procedure proposed here follows the method introduced by Hall et al. (Ann. Stat. 26, 922–942, 1998, Stat. Sin. 9, 33–49, 1999), which we modify to exploit the properties of spherical needlets. We establish convergence rates, and we show that they attain adaptivity over Besov balls in the regular region. This work is strongly motivated by issues arising in Cosmology and Astrophysics, concerning in particular the analysis of Cosmic rays.

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References

  • Autin F., Freyermuth J.-M., and von Sachs R. (2012). Combining thresholding rules: a new way to improve the performance of wavelet estimators. J. Nonparametric Stat. 24, 4, 1–18.

    MathSciNet  Google Scholar 

  • Baldi P., Kerkyacharian G., Marinucci D., and Picard D. (2009a). Asymptotics for spherical needlets. Ann. Stat. 37, 1150–1171. arXiv: http://arxiv.org/abs/math.st/0606599.

    Article  MATH  MathSciNet  Google Scholar 

  • Baldi P., Kerkyacharian G., Marinucci D., and Picard D. (2009b). Subsampling needlet coefficients on the sphere. Bernoulli 15, 438–463. arXiv: http://arxiv.org/abs/0706.4169.

    Article  MATH  MathSciNet  Google Scholar 

  • Baldi P., Kerkyacharian G., Marinucci D., and Picard D. (2009c). Adaptive density estimation for directional data using needlets. Ann. Stat. 37, 3362–3395. arXiv: http://arxiv.org/abs/0807.5059.

    Article  MATH  MathSciNet  Google Scholar 

  • Brown L.D. and Low M.G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Stat. 24, 2384–2398.

    Article  MATH  MathSciNet  Google Scholar 

  • Cabella P. and Marinucci D. (2009). Statistical challenges in the analysis of cosmic microwave background radiation. Ann. Appl. Stat. 2, 61–95.

    Article  MathSciNet  Google Scholar 

  • Cai T.T. (1999). Adaptive wavelet estimation: a block thresholding and oracle inequality approach. Ann. Stat. 27, 898–924.

    Article  MATH  Google Scholar 

  • Cai T.T. (2002). On block thresholding in wavelet regression: adaptivity, block size, and threshold level. Stat. Sin. 12, 1241–1273.

    MATH  Google Scholar 

  • Cai T.T. and Silverman B. W. (2001). Incorporating information on neighboring coefficients into wavelet estimation. Sankhya Ser. B 63, 127–148. MR1895786.

    MATH  MathSciNet  Google Scholar 

  • Cai T.T. and Zhou H.H. (2009). A data driven Block Thresholding approach to wavelet estimation.

  • Chicken E. and Cai T.T. (2005). Block thresholding for density estimation: Local and global adaptivity. J. Multivar. Anal. 95, 76–106.

    Article  MATH  MathSciNet  Google Scholar 

  • Delabrouille J., Cardoso J.-F., Le Jeune M., Betoule M., Fay G., and Guilloux F. (2008). A full sky, low foreground, high resolution CMB map from WMAP. Astron. Astrophys. 2009 493, 835–857. arXiv: http://arxiv.org/abs/0807.0773.

    Article  Google Scholar 

  • Donoho D. and Johnstone I. (1998). Minimax estimation via wavelet shrinkage. Ann. Stat. 26, 879–921.

    Article  MATH  MathSciNet  Google Scholar 

  • Donoho D., Johnstone I., Kerkyacharian G., and Picard D. (1996). Density estimation by wavelet thresholding. Ann. Stat. 24, 508–539.

    Article  MATH  MathSciNet  Google Scholar 

  • Durastanti C., Geller D., and Marinucci D. (2011). Adaptive nonparametric regression of spin fiber bundles on the sphere. J. Multivar. Anal. 104, 16–38.

    Article  MathSciNet  Google Scholar 

  • Durastanti C., Lan X., and Marinucci D. 2011a. Gaussian semiparametric estimates on the unit sphere, to be published on Bernoulli.

  • Durastanti C., Lan X., and Marinucci D. 2011b. Needlet-Whittle Estimates on the Unit Sphere, to be published on Electronic Journal of Statistics.

  • Efroimovich S.Y. (1985). Nonparametric estimation of a density of unknown smoothness. Theory Probab. Appl. 30, 557–661.

    Article  MathSciNet  Google Scholar 

  • Faÿ G., Delabrouille J., Kerkyacharian G., and Picard D. (2011). Testing the isotropy of high energy cosmic rays using spherical needlets. arXiv: http://arxiv.org/abs/1107.5658.

  • Faÿ G., Guilloux F., Betoule M., Cardoso J.-F., Delabrouille J., and Le Jeune M. (2008). CMB power spectrum estimation using wavelets. Phys. Rev. D D78, 083013. arXiv: http://arxiv.org/abs/0807.1113.

    Article  Google Scholar 

  • Geller D. and Marinucci D. (2010). Journal of fourier analysis and its applications 6, 840–884. arXiv: http://arxiv.org/abs/0811.2835.

    Article  MathSciNet  Google Scholar 

  • Geller D. and Marinucci D. (2011). Mixed needlets. J. Math. Anal. Appl. 375, 610–630.

    Article  MATH  MathSciNet  Google Scholar 

  • Geller D. and Mayeli A. (2009a). Continuous wavelets on manifolds. Math. Z. 262, 895–927. arXiv: http://arxiv.org/abs/math/0602201.

    Article  MATH  MathSciNet  Google Scholar 

  • Geller D. and Mayeli A. (2009b). Nearly tight frames and space-frequency analysis on compact manifolds. Math. Z. 263, 235–264. arXiv: http://arxiv.org/abs/0706.3642.

    Article  MATH  MathSciNet  Google Scholar 

  • Geller D. and Mayeli A. (2009c). Besov spaces and frames on compact manifolds. Indiana Univ. Math. J. 58, 2003–2042. arXiv: http://arxiv.org/abs/0709.2452.

    Article  MATH  MathSciNet  Google Scholar 

  • Hall P., Kerkyacharian G., and Picard D. (1998). Block threshold rules for curve estimation using kernel and wavelet methods. Ann. Stat. 26, 922–942.

    Article  MATH  MathSciNet  Google Scholar 

  • Hall P., Kerkyacharian G., and Picard D. (1999). On the minimax optimality of block thresholded wavelet estimators. Stat. Sin. 9, 33–49.

    MATH  MathSciNet  Google Scholar 

  • Hardle W., Kerkyacharian G., Picard D., and Tsybakov A. (1997). Wavelets, approximations and statistical application. Springer, Berlin.

    Google Scholar 

  • Iuppa R., Di Sciascio G., Hansen F.K., Marinucci D., and Santonico R. 2012. A needlet-based approach to the shower-mode data analysis in the ARGO-YBJ experiment, Nucl. Inst. Methods Phys. Res. A: Accelerators, Spectrometers, Detectors and Associated Equipment 692, 170–173.

  • Kerkyacharian G., Nickl R., and Picard D. (2010). Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds. Probab. Theory Relat. Fields (2012) 153, 363–404.

    Article  MathSciNet  Google Scholar 

  • Kerkyacharian G. and Picard D. (2004). Regression in random design and warped wavelets. Bernoulli 10, 1053–1105.

    Article  MATH  MathSciNet  Google Scholar 

  • Kim P.T. and Koo J.-Y. (2002). Optimal spherical deconvolution. J. Multivar. Anal. 80, 21–42.

    Article  MATH  MathSciNet  Google Scholar 

  • Kim P.T., Koo J.-Y., and Luo Z.-M. (2009). Weyl Eigenvalue asymptotics and sharp adaptation on vector bundles. J. Multivar. Anal. 100, 1962–1978.

    Article  MATH  MathSciNet  Google Scholar 

  • Koo J.-Y. and Kim P.T. (2008). Sharp adaptation for spherical inverse problems with applications to medical imaging. J. Multivar. Anal. 99, 165–190.

    Article  MATH  MathSciNet  Google Scholar 

  • Lan X. and Marinucci D. (2008a). The needlets bispectrum. Electron. J. Stat. 2, 332–367. arXiv: http://arxiv.org/abs/0802.4020.

    Article  MATH  MathSciNet  Google Scholar 

  • Lan X. and Marinucci D. (2008b). On the dependence structure of wavelet coefficients for spherical random fields. Stoch. Process. Appl. 119, 3749–3766. arXiv: http://arxiv.org/abs/0805.4154.

    Article  MathSciNet  Google Scholar 

  • Marinucci D. and Peccati G. 2011 Random fields on the sphere. Representation, limit theorem and cosmological applications. Cambridge University Press.

  • Marinucci D., Pietrobon D., Balbi A., Baldi P., Cabella P., Kerkyacharian G., Natoli P., Picard D., and Vittorio N. (2008). Spherical needlets for CMB data analysis. Mon. Not. R. Astron. Soc. 383, 539–545. arXiv: http://arxiv.org/abs/0707.0844.

    Article  Google Scholar 

  • Narcowich F.J., Petrushev P., and Ward J.D. (2006a). Localized tight frames on spheres. SIAM J. Math. Anal. 38, 574–594.

    Article  MathSciNet  Google Scholar 

  • Narcowich F.J., Petrushev P., and Ward J.D. (2006b). Decomposition of Besov and Triebel-Lizorkin spaces on the sphere. J. Funct. Anal. 238, 530–564.

    Article  MATH  MathSciNet  Google Scholar 

  • Nourdin I. and Peccati G. 2012. Normal approximation with Maliavin Calculus. Cambridge University Press.

  • Pietrobon D., Balbi A., and Marinucci D. (2006). Integrated Sachs-Wolfe effect from the cross correlation of WMAP3 Year and the NRAO VLA Sky Survey Data: new results and constraints on dark energy. Phys. Rev. D 74, 043524.

    Article  Google Scholar 

  • Pietrobon D., Amblard A., Balbi A., Cabella P., Cooray A., and Marinucci D. (2008). Needlet detection of features in WMAP CMB sky and the impact on anisotropies and hemispherical asymmetries. Phys. Rev. D D78, 103504. arXiv: http://arxiv.org/abs/0809.0010.

    Article  Google Scholar 

  • Pietrobon D., Cabella P., Balbi A, de Gasperis G., and Vittorio N. (2009). Constraints on primordial non-Gaussianity from a needlet analysis of the WMAP-5 data. Mon. Not. R. Astron. Soc. 396, 1682–1688. arXiv: http://arxiv.org/abs/0812.2478.

    Article  Google Scholar 

  • Rudjord O., Hansen F.K., Lan X., Liguori M., Marinucci D., and Matarrese S. (2009). An estimate of the primordial non-Gaussianity parameter f N L using the needlet bispectrum from WMAP. Astrophys. J. 701, 369–376. arXiv: http://arxiv.org/abs/0901.3154.

    Article  Google Scholar 

  • Rudjord O., Hansen F.K., Lan X., Liguori M., Marinucci D., and Matarrese S. (2010). Directional variations of the non-Gaussianity parameter f N L . Astrophys. J. 708, 1321–1325. arXiv: http://arxiv.org/abs/0906.3232.

    Article  Google Scholar 

  • Tsybakov A.B. (2009). Introduction to nonparametric estimation. Springer, New York.

    Book  MATH  Google Scholar 

  • Varshalovich D.A., Moskalev A.N., and Khersonskii V.K. (1988). Quantum theory of angular momentum. World Scientific, Singapore.

    Book  Google Scholar 

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Authors and Affiliations

  1. University of “Tor Vergata”, Rome, Italy

    Claudio Durastanti

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  1. Claudio Durastanti
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Correspondence to Claudio Durastanti.

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Research supported by ERC Grant n. 277742 Pascal

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Durastanti, C. Block Thresholding on the Sphere. Sankhya A 77, 153–185 (2015). https://doi.org/10.1007/s13171-014-0057-0

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  • DOI: https://doi.org/10.1007/s13171-014-0057-0

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