IDEAS home Printed from https://ideas.repec.org/a/bla/scjsta/v36y2009i4p749-763.html

Log‐density Deconvolution by Wavelet Thresholding

Author

Listed:
  • JÉRÉMIE BIGOT
  • SÉBASTIEN VAN BELLEGEM

Abstract

. This paper proposes a new wavelet‐based method for deconvolving a density. The estimator combines the ideas of non‐linear wavelet thresholding with periodized Meyer wavelets and estimation by information projection. It is guaranteed to be in the class of density functions, in particular it is positive everywhere by construction. The asymptotic optimality of the estimator is established in terms of the rate of convergence of the Kullback–Leibler discrepancy over Besov classes. Finite sample properties are investigated in detail, and show the excellent empirical performance of the estimator, compared with other recently introduced estimators.

Suggested Citation

  • Jérémie Bigot & Sébastien Van Bellegem, 2009. "Log‐density Deconvolution by Wavelet Thresholding," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(4), pages 749-763, December.
    • Handle: RePEc:bla:scjsta:v:36:y:2009:i:4:p:749-763
    DOI: 10.1111/j.1467-9469.2009.00653.x
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/j.1467-9469.2009.00653.x
    Download Restriction: no
    File URL: https://libkey.io/10.1111/j.1467-9469.2009.00653.x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Koo, Ja-Yong & Kim, Woo-Chul, 1996. "Wavelet density estimation by approximation of log-densities," Statistics & Probability Letters, Elsevier, vol. 26(3), pages 271-278, February.
    2. Peter Hall & Peihua Qiu, 2005. "Discrete-transform approach to deconvolution problems," Biometrika, Biometrika Trust, vol. 92(1), pages 135-148, March.
    3. JOHANNES, Jan & VAN BELLEGHEM, Sébastien & VANHEMS, Anne, 2007. "A unified approach to solve ill-posed inverse problems in econometrics," LIDAM Discussion Papers CORE 2007083, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Johannes, Jan & Van Bellegem, Sébastien & Vanhems, Anne, 2011. "Convergence Rates For Ill-Posed Inverse Problems With An Unknown Operator," Econometric Theory, Cambridge University Press, vol. 27(3), pages 522-545, June.
    5. Ja‐Yong Koo, 1999. "Logspline Deconvolution in Besov Space," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(1), pages 73-86, March.
    6. Daniela De Canditiis & Marianna Pensky, 2006. "Simultaneous Wavelet Deconvolution in Periodic Setting," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 293-306, June.
    7. Iain M. Johnstone & Gérard Kerkyacharian & Dominique Picard & Marc Raimondo, 2004. "Wavelet deconvolution in a periodic setting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(3), pages 547-573, August.
    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. Florens, Jean-Pierre & Schwarz, Maik & Van Bellegem, Sébastien, 2010. "Nonparametric Frontier Estimation from Noisy Data," TSE Working Papers 10-179, Toulouse School of Economics (TSE).
    2. Bak, Kwan-Young & Jhong, Jae-Hwan & Lee, JungJun & Shin, Jae-Kyung & Koo, Ja-Yong, 2021. "Penalized logspline density estimation using total variation penalty," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).
    3. Johannes, Jan & Van Bellegem, Sébastien & Vanhems, Anne, 2011. "Convergence Rates For Ill-Posed Inverse Problems With An Unknown Operator," Econometric Theory, Cambridge University Press, vol. 27(3), pages 522-545, June.

    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. Petsa, Athanasia & Sapatinas, Theofanis, 2009. "Minimax convergence rates under the Lp-risk in the functional deconvolution model," Statistics & Probability Letters, Elsevier, vol. 79(13), pages 1568-1576, July.
    2. Bak, Kwan-Young & Jhong, Jae-Hwan & Lee, JungJun & Shin, Jae-Kyung & Koo, Ja-Yong, 2021. "Penalized logspline density estimation using total variation penalty," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).
    3. 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.
    4. Bissantz, Nicolai & Holzmann, Hajo, 2007. "Statistical inference for inverse problems," Technical Reports 2007,40, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    5. Jad Beyhum & Elia Lapenta & Pascal Lavergne, 2025. "One-step smoothing splines instrumental regression," The Econometrics Journal, Royal Economic Society, vol. 28(2), pages 176-197.
    6. Daniel Wilhelm, 2015. "Identification and estimation of nonparametric panel data regressions with measurement error," CeMMAP working papers 34/15, Institute for Fiscal Studies.
    7. William Horrace & Christopher Parmeter, 2011. "Semiparametric deconvolution with unknown error variance," Journal of Productivity Analysis, Springer, vol. 35(2), pages 129-141, April.
    8. Raymond J. Carroll & Aurore Delaigle & Peter Hall, 2007. "Non‐parametric regression estimation from data contaminated by a mixture of Berkson and classical errors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(5), pages 859-878, November.
    9. Schwarz, Maik & Van Bellegem, Sébastien, 2010. "Consistent density deconvolution under partially known error distribution," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 236-241, February.
    10. Jad Beyhum & Elia Lapenta & Pascal Lavergne, 2023. "One-step nonparametric instrumental regression using smoothing splines," Working Papers hal-04971401, HAL.
    11. Daniel Wilhelm, 2015. "Identification and estimation of nonparametric panel data regressions with measurement error," CeMMAP working papers CWP34/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    12. Melanie Birke & Sebastien Van Bellegem & Ingrid Van Keilegom, 2017. "Semi-parametric Estimation in a Single-index Model with Endogenous Variables," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(1), pages 168-191, March.
    13. Feve, Frederique & Florens, Jean-Pierre & Van Keilegom, Ingrid, 2012. "Estimation of conditional ranks and tests of exogeneity in nonparametric nonseparable models," LIDAM Discussion Papers ISBA 2012036, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    14. Johannes, Jan & Van Bellegem, Sébastien & Vanhems, Anne, 2010. "Iterative Regularization in Nonparametric Instrumental Regression," TSE Working Papers 10-184, Toulouse School of Economics (TSE).
    15. Frédérique Fève & Jean-Pierre Florens, 2010. "The practice of non-parametric estimation by solving inverse problems: the example of transformation models," Econometrics Journal, Royal Economic Society, vol. 13(3), pages 1-27, October.
    16. Florens, Jean-Pierre & Van Bellegem, Sébastien, 2015. "Instrumental variable estimation in functional linear models," Journal of Econometrics, Elsevier, vol. 186(2), pages 465-476.
    17. Koo, Ja-Yong & Kooperberg, Charles, 2000. "Logspline density estimation for binned data," Statistics & Probability Letters, Elsevier, vol. 46(2), pages 133-147, January.
    18. Florens, Jean-Pierre & Schwarz, Maik & Van Bellegem, Sébastien, 2010. "Nonparametric Frontier Estimation from Noisy Data," IDEI Working Papers 625, Institut d'Économie Industrielle (IDEI), Toulouse.
    19. Martin L. Hazelton & Berwin A. Turlach, 2010. "Semiparametric Density Deconvolution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(1), pages 91-108, March.
    20. Nicolai Bissantz & Gerda Claeskens & Hajo Holzmann & Axel Munk, 2009. "Testing for lack of fit in inverse regression—with applications to biophotonic imaging," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 25-48, January.

    More about this item

    Statistics

    Access and download statistics

    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:bla:scjsta:v:36:y:2009:i:4:p:749-763. 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: Wiley Content Delivery (email available below). General contact details of provider: http://www.blackwellpublishing.com/journal.asp?ref=0303-6898 .

    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.