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Adaptive global thresholding on the sphere

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  • Durastanti, Claudio

Abstract

This work is concerned with the study of the adaptivity properties of nonparametric regression estimators over the d-dimensional sphere within the global thresholding framework. The estimators are constructed by means of a form of spherical wavelets, the so-called needlets, which enjoy strong concentration properties in both harmonic and real domains. The author establishes the convergence rates of the Lp-risks of these estimators, focusing on their minimax properties and proving their optimality over a scale of nonparametric regularity function spaces, namely, the Besov spaces.

Suggested Citation

  • Durastanti, Claudio, 2016. "Adaptive global thresholding on the sphere," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 110-132.
    • Handle: RePEc:eee:jmvana:v:151:y:2016:i:c:p:110-132
    DOI: 10.1016/j.jmva.2016.07.009
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    References listed on IDEAS

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    1. Durastanti, Claudio & Geller, Daryl & Marinucci, Domenico, 2012. "Adaptive nonparametric regression on spin fiber bundles," Journal of Multivariate Analysis, Elsevier, vol. 104(1), pages 16-38, February.
    2. Eric Gautier & Erwann Le Pennec, 2011. "Adaptive Estimation in the Nonparametric Random Coefficients Binary Choice Model by Needlet Thresholding," Working Papers 2011-20, Center for Research in Economics and Statistics.
    3. Kerkyacharian, G. & Picard, D., 1992. "Density estimation in Besov spaces," Statistics & Probability Letters, Elsevier, vol. 13(1), pages 15-24, January.
    4. Kim, Peter T. & Koo, Ja-Yong, 2002. "Optimal Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 21-42, January.
    5. Gérard Kerkyacharian & Dominique Picard & Lucien Birgé & Peter Hall & Oleg Lepski & Enno Mammen & Alexandre Tsybakov & G. Kerkyacharian & D. Picard, 2000. "Thresholding algorithms, maxisets and well-concentrated bases," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(2), pages 283-344, December.
    6. Jean-Baptiste Monnier, 2011. "Nonparametric regression on the hyper-sphere with uniform design," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 412-446, August.
    7. Koo, Ja-Yong & Kim, Peter T., 2008. "Sharp adaptation for spherical inverse problems with applications to medical imaging," Journal of Multivariate Analysis, Elsevier, vol. 99(2), pages 165-190, February.
    8. Kerkyacharian, Gérard & Picard, Dominique, 1993. "Density estimation by kernel and wavelets methods: Optimality of Besov spaces," Statistics & Probability Letters, Elsevier, vol. 18(4), pages 327-336, November.
    9. Lan, Xiaohong & Marinucci, Domenico, 2009. "On the dependence structure of wavelet coefficients for spherical random fields," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3749-3766, October.
    10. Kim, Peter T. & Koo, Ja-Yong & Luo, Zhi-Ming, 2009. "Weyl eigenvalue asymptotics and sharp adaptation on vector bundles," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1962-1978, October.
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    Cited by:

    1. Maurizia Rossi, 2019. "The Defect of Random Hyperspherical Harmonics," Journal of Theoretical Probability, Springer, vol. 32(4), pages 2135-2165, December.

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