IDEAS home Printed from https://ideas.repec.org/a/bla/scjsta/v36y2009i4p749-763.html
   My bibliography  Save this article

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. 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.
    2. 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).
    3. 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.
    4. 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.
    5. Peter Hall & Peihua Qiu, 2005. "Discrete-transform approach to deconvolution problems," Biometrika, Biometrika Trust, vol. 92(1), pages 135-148, March.
    6. 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.
    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. Bissantz, Nicolai & Holzmann, Hajo, 2007. "Statistical inference for inverse problems," Technical Reports 2007,40, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    3. 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.
    4. S. Darolles & Y. Fan & J. P. Florens & E. Renault, 2011. "Nonparametric Instrumental Regression," Econometrica, Econometric Society, vol. 79(5), pages 1541-1565, September.
    5. 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.
    6. 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).
    7. Florens, Jean-Pierre & Simoni, Anna, 2016. "Regularizing Priors For Linear Inverse Problems," Econometric Theory, Cambridge University Press, vol. 32(1), pages 71-121, February.
    8. 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.
    9. Koo, Ja-Yong & Kooperberg, Charles, 2000. "Logspline density estimation for binned data," Statistics & Probability Letters, Elsevier, vol. 46(2), pages 133-147, January.
    10. 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.
    11. Chesneau, Christophe, 2007. "Regression with random design: A minimax study," Statistics & Probability Letters, Elsevier, vol. 77(1), pages 40-53, January.
    12. Abdous, Belkacem & Germain, Stéphane & Ghazzali, Nadia, 2002. "A unified treatment of direct and indirect estimation of a probability density and its derivatives," Statistics & Probability Letters, Elsevier, vol. 56(3), pages 239-250, February.
    13. J. Andrés Christen & Bruno Sansó & Mario Santana-Cibrian & Jorge X. Velasco-Hernández, 2016. "Bayesian deconvolution of oil well test data using Gaussian processes," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(4), pages 721-737, March.
    14. Raimondo, Marc & Stewart, Michael, 2007. "The WaveD Transform in R: Performs Fast Translation-Invariant Wavelet Deconvolution," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 21(i02).
    15. Nicolai Bissantz & Hajo Holzmann, 2013. "Asymptotics for spectral regularization estimators in statistical inverse problems," Computational Statistics, Springer, vol. 28(2), pages 435-453, April.
    16. Abbaszadeh, Mohammad & Chesneau, Christophe & Doosti, Hassan, 2012. "Nonparametric estimation of density under bias and multiplicative censoring via wavelet methods," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 932-941.
    17. Maarten Jansen & Guy P. Nason & B. W. Silverman, 2009. "Multiscale methods for data on graphs and irregular multidimensional situations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 97-125, January.
    18. Birke, Melanie & Bissantz, Nicolai, 2007. "Shape constrained estimators in inverse regression models with convolution-type operator," Technical Reports 2007,35, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    19. Koo, Ja-Yong, 1998. "Convergence Rates for Logspline Tomography," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 367-384, November.
    20. Fève, Frédérique & Florens, Jean-Pierre, 2014. "Non parametric analysis of panel data models with endogenous variables," Journal of Econometrics, Elsevier, vol. 181(2), pages 151-164.

    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.