IDEAS home Printed from
   My bibliography  Save this article

Halfplane Trimming for Bivariate Distributions


  • Masse, J. C.
  • Theodorescu, R.


Let [mu] be a probability measure on R2 and let u [set membership, variant] (0, 1). A bivariate u-trimmed region D(u), defined as the intersection of all halfplanes whose [mu]-probability measure is at least equal to u, is studied. It is shown that D(u) is not empty for u sufficiently close to 1 and that D(u) satisfies some natural continuity properties. Limit behavior is also considered, the main result being that the weak convergence of a sequence of probability measures entails the pointwise convergence with respect to Hausdorff distance of the associated trimmed regions; this is then applied to derive asymptotics of the empirical trimmed regions. A brief discussion of the extension of the results to higher dimensions is also given.

Suggested Citation

  • Masse, J. C. & Theodorescu, R., 1994. "Halfplane Trimming for Bivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 48(2), pages 188-202, February.
  • Handle: RePEc:eee:jmvana:v:48:y:1994:i:2:p:188-202

    Download full text from publisher

    File URL:
    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.


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

    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. 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. Massé, Jean-Claude, 2002. "Asymptotics for the Tukey Median," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 286-300, May.
    4. Romanazzi, Mario, 2001. "Influence Function of Halfspace Depth," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 138-161, April.
    5. 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.
    6. McNeil, Alexander J. & Smith, Andrew D., 2012. "Multivariate stress scenarios and solvency," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 299-308.
    7. Barme-Delcroix, Marie-Francoise & Gather, Ursula, 2007. "Limit laws for multidimensional extremes," Statistics & Probability Letters, Elsevier, vol. 77(18), pages 1750-1755, December.
    8. Ruts, Ida & Rousseeuw, Peter J., 1996. "Computing depth contours of bivariate point clouds," Computational Statistics & Data Analysis, Elsevier, vol. 23(1), pages 153-168, November.
    9. Cousin, Areski & Di Bernardino, Elena, 2014. "On multivariate extensions of Conditional-Tail-Expectation," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 272-282.
    10. 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.
    11. Ignacio Cascos & Ilya Molchanov, 2006. "Multivariate risks and depth-trimmed regions," Papers math/0606520,, revised Nov 2006.
    12. Kong, Linglong & Zuo, Yijun, 2010. "Smooth depth contours characterize the underlying distribution," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2222-2226, October.
    13. Areski Cousin & Elena Di Bernadino, 2011. "On Multivariate Extensions of Value-at-Risk," Papers 1111.1349,, revised Apr 2013.
    14. repec:spr:advdac:v:11:y:2017:i:3:d:10.1007_s11634-016-0269-3 is not listed on IDEAS
    15. Areski Cousin & Elena Di Bernadino, 2013. "On Multivariate Extensions of Value-at-Risk," Working Papers hal-00638382, HAL.
    16. Cascos, Ignacio & López-Díaz, Miguel, 2005. "Integral trimmed regions," Journal of Multivariate Analysis, Elsevier, vol. 96(2), pages 404-424, October.
    17. Cousin, Areski & Di Bernardino, Elena, 2013. "On multivariate extensions of Value-at-Risk," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 32-46.
    18. Kuelbs, James & Zinn, Joel, 2016. "Convergence of quantile and depth regions," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3681-3700.
    19. 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.
    20. Belzunce, F. & Castano, A. & Olvera-Cervantes, A. & Suarez-Llorens, A., 2007. "Quantile curves and dependence structure for bivariate distributions," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 5112-5129, June.
    21. Areski Cousin & Elena Di Bernardinoy, 2013. "On Multivariate Extensions of Conditional-Tail-Expectation," Working Papers hal-00877386, HAL.
    22. Cascos, Ignacio, 2006. "The expected convex hull trimmed regions of a sample," DES - Working Papers. Statistics and Econometrics. WS ws066919, Universidad Carlos III de Madrid. Departamento de Estadística.
    23. Elena Di Bernardino & Thomas Laloë & Véronique Maume-Deschamps & Clémentine Prieur, 2013. "Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory," Post-Print hal-00580624, HAL.

    More about this item


    Access and download statistics


    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:48:y:1994:i:2:p:188-202. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dana Niculescu). General contact details of provider: .

    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.

    We have no references for this item. You can help adding them by using 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.