IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v39y1998i2p179-184.html
   My bibliography  Save this article

Convergence in the Hausdorff metric of estimators of irregular densities, using Fourier-Cesàro approximation

Author

Listed:
  • van Rooij, Arnoud C. M.
  • Ruymgaart, Frits H.

Abstract

The problem of estimating a density which is allowed to have discontinuities of the first kind is considered. The usual Fourier-type estimator is based on the Dirichlet or sine kernel and is not suitable to eliminate the Gibbs phenomenon. Fourier-Cesàro approximation yields the Fejér kernel which is the square of the sine function. Density estimators based on the Fejér kernel do control the Gibbs phenomenon. Integral metrics are not sufficiently sensitive to properly assess the performance of estimators of irregular signals. Therefore, we use the Hausdorff distance between the extended, closed, graphs of estimator and estimand, and derive an a.s. speed of convergence of this distance.

Suggested Citation

  • van Rooij, Arnoud C. M. & Ruymgaart, Frits H., 1998. "Convergence in the Hausdorff metric of estimators of irregular densities, using Fourier-Cesàro approximation," Statistics & Probability Letters, Elsevier, vol. 39(2), pages 179-184, August.
    • Handle: RePEc:eee:stapro:v:39:y:1998:i:2:p:179-184
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-7152(98)00062-5
    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. Kerkyacharian, G. & Picard, D., 1992. "Density estimation in Besov spaces," Statistics & Probability Letters, Elsevier, vol. 13(1), pages 15-24, January.
    Full references (including those not matched with items on IDEAS)

    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. Kato, Takeshi, 1999. "Density estimation by truncated wavelet expansion," Statistics & Probability Letters, Elsevier, vol. 43(2), pages 159-168, June.
    2. Gérard, Kerkyacharian & Dominique, Picard, 1997. "Limit of the quadratic risk in density estimation using linear methods," Statistics & Probability Letters, Elsevier, vol. 31(4), pages 299-312, February.
    3. Karun Adusumilli & Taisuke Otsu, 2015. "Nonparametric instrumental regression with errors in variables," STICERD - Econometrics Paper Series /2015/585, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    4. A. Antoniadis, 1997. "Wavelets in statistics: A review," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 6(2), pages 97-130, August.
    5. Masry, Elias, 1997. "Multivariate probability density estimation by wavelet methods: Strong consistency and rates for stationary time series," Stochastic Processes and their Applications, Elsevier, vol. 67(2), pages 177-193, May.
    6. Durastanti, Claudio, 2016. "Adaptive global thresholding on the sphere," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 110-132.
    7. Antonio Cosma & Olivier Scaillet & Rainer von Sachs, 2005. "Multiariate Wavelet-based sahpe preserving estimation for dependant observation," FAME Research Paper Series rp144, International Center for Financial Asset Management and Engineering.
    8. García Treviño, E.S. & Alarcón Aquino, V. & Barria, J.A., 2019. "The radial wavelet frame density estimator," Computational Statistics & Data Analysis, Elsevier, vol. 130(C), pages 111-139.
    9. Zeng, Xiaochen & Wang, Jinru, 2018. "Wavelet density deconvolution estimations with heteroscedastic measurement errors," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 79-85.
    10. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2011. "Inference on Treatment Effects After Selection Amongst High-Dimensional Controls," Papers 1201.0224, arXiv.org, revised May 2012.
    11. Gaëlle Chagny & Claire Lacour, 2015. "Optimal adaptive estimation of the relative density," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 605-631, September.
    12. Autin, F. & Le Pennec, E. & Tribouley, K., 2010. "Thresholding methods to estimate copula density," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 200-222, January.
    13. Hoffmann, Marc, 1997. "Minimax estimation of the diffusion coefficient through irregular samplings," Statistics & Probability Letters, Elsevier, vol. 32(1), pages 11-24, February.
    14. Liang, Han-Ying & de Uña-Álvarez, Jacobo, 2011. "Wavelet estimation of conditional density with truncated, censored and dependent data," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 448-467, March.
    15. Marianna Pensky, 2002. "Locally Adaptive Wavelet Empirical Bayes Estimation of a Location Parameter," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(1), pages 83-99, March.
    16. 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.
    17. Leblanc, Frédérique, 1996. "Wavelet linear density estimator for a discrete-time stochastic process: Lp-losses," Statistics & Probability Letters, Elsevier, vol. 27(1), pages 71-84, March.
    18. Naono Ken, 1995. "Comparative Computations of Non-parametric Density Estimation Between Some Kernel Method and the Wavelet Method," Monte Carlo Methods and Applications, De Gruyter, vol. 1(2), pages 147-163, December.
    19. Sultana Didi & Salim Bouzebda, 2022. "Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes," Mathematics, MDPI, vol. 10(22), pages 1-37, November.
    20. Genest, Christian & Masiello, Esterina & Tribouley, Karine, 2009. "Estimating copula densities through wavelets," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 170-181, April.

    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:stapro:v:39:y:1998:i:2:p:179-184. 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.