IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v100y2009i9p1962-1978.html
   My bibliography  Save this article

Weyl eigenvalue asymptotics and sharp adaptation on vector bundles

Author

Listed:
  • Kim, Peter T.
  • Koo, Ja-Yong
  • Luo, Zhi-Ming

Abstract

This paper examines the estimation of an indirect signal embedded in white noise on vector bundles. It is found that the sharp asymptotic minimax bound is determined by the degree to which the indirect signal is embedded in the linear operator. Thus when the linear operator has polynomial decay, recovery of the signal is polynomial where the exact minimax constant and rate are determined. Adaptive sharp estimation is carried out using a blockwise shrinkage estimator. Application to the spherical deconvolution problem for the polynomially bounded case is made.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:100:y:2009:i:9:p:1962-1978
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(09)00064-5
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Healy, Dennis M. & Hendriks, Harrie & Kim, Peter T., 1998. "Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 67(1), pages 1-22, October.
    2. Kim, Peter T. & Koo, Ja-Yong & Park, Heon Jin, 2004. "Sharp minimaxity and spherical deconvolution for super-smooth error distributions," Journal of Multivariate Analysis, Elsevier, vol. 90(2), pages 384-392, August.
    3. Kim, Peter T. & Koo, Ja-Yong, 2002. "Optimal Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 21-42, January.
    4. Jussi Klemela, 1999. "Asymptotic Minimax Risk for the White Noise Model on the Sphere," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(3), pages 465-473, September.
    5. 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.
    6. Bissantz, Nicolai & Hohage, T. & Munk, Axel & Ruymgaart, F., 2007. "Convergence rates of general regularization methods for statistical inverse problems and applications," Technical Reports 2007,04, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    7. Goldenshluger, Alexander, 2002. "Density Deconvolution in the Circular Structural Model," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 360-375, May.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Durastanti, Claudio, 2016. "Adaptive global thresholding on the sphere," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 110-132.
    2. Hielscher, Ralf, 2013. "Kernel density estimation on the rotation group and its application to crystallographic texture analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 119-143.
    3. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.
    2. Kim, Peter T. & Koo, Ja-Yong & Park, Heon Jin, 2004. "Sharp minimaxity and spherical deconvolution for super-smooth error distributions," Journal of Multivariate Analysis, Elsevier, vol. 90(2), pages 384-392, August.
    3. 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.
    4. Jammalamadaka, S. Rao & Terdik, György H., 2019. "Harmonic analysis and distribution-free inference for spherical distributions," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 436-451.
    5. Crane, M. & Patrangenaru, V., 2011. "Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair images," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 225-237, February.
    6. Vareschi, T., 2014. "Application of second generation wavelets to blind spherical deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 398-417.
    7. Durastanti, Claudio, 2016. "Adaptive global thresholding on the sphere," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 110-132.
    8. Bissantz, Nicolai & Hohage, T. & Munk, Axel & Ruymgaart, F., 2007. "Convergence rates of general regularization methods for statistical inverse problems and applications," Technical Reports 2007,04, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    9. Pham Ngoc, Thanh Mai & Rivoirard, Vincent, 2013. "The dictionary approach for spherical deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 138-156.
    10. 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.
    11. Hendriks, Harrie, 2003. "Application of fast spherical Fourier transform to density estimation," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 209-221, February.
    12. Kim, Yoon Tae & Park, Hyun Suk, 2013. "Geometric structures arising from kernel density estimation on Riemannian manifolds," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 112-126.
    13. Florens, Jean-Pierre & Simoni, Anna, 2016. "Regularizing Priors For Linear Inverse Problems," Econometric Theory, Cambridge University Press, vol. 32(1), pages 71-121, February.
    14. Raymond Carroll & Xiaohong Chen & Yingyao Hu, 2010. "Identification and estimation of nonlinear models using two samples with nonclassical measurement errors," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(4), pages 419-423.
    15. Christian Wagner & Ulrich Stadtmüller, 2008. "Asymptotics for TAYLEX and SIMEX estimators in deconvolution of densities," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(6), pages 507-522.
    16. Matthew Thorpe & Adam M. Johansen, 2018. "Pointwise convergence in probability of general smoothing splines," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 717-744, August.
    17. Xiaohong Chen & Demian Pouzo, 2008. "Estimation of nonparametric conditional moment models with possibly nonsmooth moments," CeMMAP working papers CWP12/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    18. Hoderlein, Stefan & Nesheim, Lars & Simoni, Anna, 2017. "Semiparametric Estimation Of Random Coefficients In Structural Economic Models," Econometric Theory, Cambridge University Press, vol. 33(6), pages 1265-1305, December.
    19. Marteau Clement & Loubes Jean-Michel, 2012. "Adaptive estimation for an inverse regression model with unknown operator," Statistics & Risk Modeling, De Gruyter, vol. 29(3), pages 215-242, August.
    20. Colin Griesbach & Andreas Mayr & Elisabeth Bergherr, 2023. "Variable Selection and Allocation in Joint Models via Gradient Boosting Techniques," Mathematics, MDPI, vol. 11(2), pages 1-16, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:100:y:2009:i:9:p:1962-1978. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.