IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v83y2002i2p360-364.html
   My bibliography  Save this article

The Tukey Depth Characterizes the Atomic Measure

Author

Listed:
  • Koshevoy, Gleb A.

Abstract

We prove that if the Tukey depths of two atomic measures are identical, then the measures are also identical.

Suggested Citation

  • Koshevoy, Gleb A., 2002. "The Tukey Depth Characterizes the Atomic Measure," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 360-364, November.
    • Handle: RePEc:eee:jmvana:v:83:y:2002:i:2:p:360-364
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(01)92052-4
    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. Nolan, D., 1992. "Asymptotics for multivariate trimming," Stochastic Processes and their Applications, Elsevier, vol. 42(1), pages 157-169, August.
    2. Struyf, Anja J. & Rousseeuw, Peter J., 1999. "Halfspace Depth and Regression Depth Characterize the Empirical Distribution," Journal of Multivariate Analysis, Elsevier, vol. 69(1), pages 135-153, April.
    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. Hassairi, Abdelhamid & Regaieg, Ons, 2008. "On the Tukey depth of a continuous probability distribution," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2308-2313, October.
    2. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    3. Stanislav Nagy, 2021. "Halfspace depth does not characterize probability distributions," Statistical Papers, Springer, vol. 62(3), pages 1135-1139, June.
    4. Wei, Bei & Lee, Stephen M.S., 2012. "Second-order accuracy of depth-based bootstrap confidence regions," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 112-123.
    5. repec:hal:spmain:info:hdl:2441/64itsev5509q8aa5mrbhi0g0b6 is not listed on IDEAS
    6. Dyckerhoff, Rainer & Mozharovskyi, Pavlo, 2016. "Exact computation of the halfspace depth," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 19-30.
    7. repec:hal:spmain:info:hdl:2441/3qnaslliat80pbqa8t90240unj is not listed on IDEAS
    8. Kong, Linglong & Zuo, Yijun, 2010. "Smooth depth contours characterize the underlying distribution," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2222-2226, October.
    9. repec:spo:wpmain:info:hdl:2441/64itsev5509q8aa5mrbhi0g0b6 is not listed on IDEAS
    10. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    11. Laketa, Petra & Nagy, Stanislav, 2021. "Reconstruction of atomic measures from their halfspace depth," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    12. Xiaohui Liu & Karl Mosler & Pavlo Mozharovskyi, 2017. "Fast computation of Tukey trimmed regions and median in dimension p > 2," Working Papers 2017-71, Center for Research in Economics and Statistics.
    13. Cuesta-Albertos, J.A. & Nieto-Reyes, A., 2008. "The Tukey and the random Tukey depths characterize discrete distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2304-2311, November.

    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. Petra Laketa & Stanislav Nagy, 2022. "Halfspace depth for general measures: the ray basis theorem and its consequences," Statistical Papers, Springer, vol. 63(3), pages 849-883, June.
    2. Mia Hubert & Peter Rousseeuw & Pieter Segaert, 2015. "Multivariate functional outlier detection," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 177-202, July.
    3. Gather, Ursula & Fried, Roland & Lanius, Vivian, 2005. "Robust detail-preserving signal extraction," Technical Reports 2005,54, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    4. Averous, Jean & Meste, Michel, 1997. "Median Balls: An Extension of the Interquantile Intervals to Multivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 63(2), pages 222-241, November.
    5. Jonas Baillien & Irène Gijbels & Anneleen Verhasselt, 2023. "Flexible asymmetric multivariate distributions based on two-piece univariate distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(1), pages 159-200, February.
    6. Christmann, Andreas & Steinwart, Ingo & Hubert, Mia, 2006. "Robust Learning from Bites for Data Mining," Technical Reports 2006,03, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    7. Christmann, Andreas & Steinwart, Ingo, 2005. "Consistency and robustness of kernel based regression," Technical Reports 2005,01, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    8. Zuo, Yijun & Serfling, Robert, 2000. "Nonparametric Notions of Multivariate "Scatter Measure" and "More Scattered" Based on Statistical Depth Functions," Journal of Multivariate Analysis, Elsevier, vol. 75(1), pages 62-78, October.
    9. Stanislav Nagy, 2021. "Halfspace depth does not characterize probability distributions," Statistical Papers, Springer, vol. 62(3), pages 1135-1139, June.
    10. Yi He & John H. J. Einmahl, 2017. "Estimation of extreme depth-based quantile regions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 449-461, March.
    11. Polonik, Wolfgang, 1997. "Minimum volume sets and generalized quantile processes," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 1-24, July.
    12. Kim, Jeankyung, 2000. "Rate of convergence of depth contours: with application to a multivariate metrically trimmed mean," Statistics & Probability Letters, Elsevier, vol. 49(4), pages 393-400, October.
    13. Nolan, D., 1999. "On min-max majority and deepest points," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 325-333, July.
    14. Xiaohui Liu & Karl Mosler & Pavlo Mozharovskyi, 2017. "Fast computation of Tukey trimmed regions and median in dimension p > 2," Working Papers 2017-71, Center for Research in Economics and Statistics.
    15. Hassairi, Abdelhamid & Regaieg, Ons, 2008. "On the Tukey depth of a continuous probability distribution," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2308-2313, October.
    16. Wang, Jin, 2018. "A note on weak convergence of general halfspace depth trimmed means," Statistics & Probability Letters, Elsevier, vol. 142(C), pages 50-56.
    17. Christmann, Andreas & Steinwart, Ingo, 2003. "On robustness properties of convex risk minimization methods for pattern recognition," Technical Reports 2003,15, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    18. Christmann, Andreas, 2004. "Regression depth and support vector machine," Technical Reports 2004,54, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    19. Wang, Jin, 2019. "Asymptotics of generalized depth-based spread processes and applications," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 363-380.
    20. Giorgi, Emanuele & McNeil, Alexander J., 2016. "On the computation of multivariate scenario sets for the skew-t and generalized hyperbolic families," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 205-220.

    More about this item

    Keywords

    halfspace depth depth contour;

    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:eee:jmvana:v:83:y:2002:i:2:p:360-364. 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.