IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v77y2001i1p138-161.html
   My bibliography  Save this article

Influence Function of Halfspace Depth

Author

Listed:
  • Romanazzi, Mario

Abstract

The sensitivity of halfspace depth values and contours to perturbations of the underlying distribution is investigated. The influence function of the halfspace depth of any point x[set membership, variant]p is bounded and discontinuous; it is constant and positive when the perturbing observation z is placed in any optimal halfspace and it is constant and negative when z is placed in any non-optimal halfspace. When the optimal halfspace is unique a von Mises expansion allows an easy derivation of the asymptotic distribution of the sample halfspace depth. In the sampling case, in general, addition of a single observation outside the convex hull of the sample alters all the depth regions but only the outer region can be arbitrarily expanded. To obtain the same effect on the inner regions the size of the perturbation is required to be not less than the depth orders. Numerical illustrations of the results are given.

Suggested Citation

  • Romanazzi, Mario, 2001. "Influence Function of Halfspace Depth," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 138-161, April.
    • Handle: RePEc:eee:jmvana:v:77:y:2001:i:1:p:138-161
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(00)91929-8
    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. 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.
    3. Chen, Z. Q., 1995. "Bounds for the Breakdown Point of the Simplicial Median," Journal of Multivariate Analysis, Elsevier, vol. 55(1), pages 1-13, October.
    4. Masse, J. C. & Theodorescu, R., 1994. "Halfplane Trimming for Bivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 48(2), pages 188-202, February.
    5. Nolan, D., 1999. "On min-max majority and deepest points," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 325-333, July.
    6. Peter J. Rousseeuw & Ida Ruts, 1996. "Bivariate Location Depth," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 45(4), pages 516-526, December.
    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. 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.
    2. Benjamin Avanzi & Mark Lavender & Greg Taylor & Bernard Wong, 2022. "Detection and treatment of outliers for multivariate robust loss reserving," Papers 2203.03874, arXiv.org, revised Jun 2023.
    3. Wang, Jin, 2018. "A note on weak convergence of general halfspace depth trimmed means," Statistics & Probability Letters, Elsevier, vol. 142(C), pages 50-56.
    4. Xin Dang & Robert Serfling & Weihua Zhou, 2009. "Influence functions of some depth functions, and application to depth-weighted L-statistics," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(1), pages 49-66.
    5. Wang, Jin & Serfling, Robert, 2006. "Influence functions for a general class of depth-based generalized quantile functions," Journal of Multivariate Analysis, Elsevier, vol. 97(4), pages 810-826, April.

    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. 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.
    2. Mia Hubert & Peter Rousseeuw & Pieter Segaert, 2017. "Multivariate and functional classification using depth and distance," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 11(3), pages 445-466, September.
    3. 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.
    4. 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.
    5. repec:cte:wsrepe:28434 is not listed on IDEAS
    6. 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.
    7. Dyckerhoff, Rainer & Mozharovskyi, Pavlo, 2016. "Exact computation of the halfspace depth," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 19-30.
    8. Nolan, D., 1999. "On min-max majority and deepest points," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 325-333, July.
    9. Hamel, Andreas H. & Kostner, Daniel, 2022. "Computation of quantile sets for bivariate ordered data," Computational Statistics & Data Analysis, Elsevier, vol. 169(C).
    10. repec:cte:wsrepe:35465 is not listed on IDEAS
    11. 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.
    12. Chakraborty, Biman & Chaudhuri, Probal, 1999. "A note on the robustness of multivariate medians," Statistics & Probability Letters, Elsevier, vol. 45(3), pages 269-276, November.
    13. 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.
    14. Struyf, Anja & Rousseeuw, Peter J., 2000. "High-dimensional computation of the deepest location," Computational Statistics & Data Analysis, Elsevier, vol. 34(4), pages 415-426, October.
    15. Mosler, Karl & Lange, Tatjana & Bazovkin, Pavel, 2009. "Computing zonoid trimmed regions of dimension d>2," Computational Statistics & Data Analysis, Elsevier, vol. 53(7), pages 2500-2510, May.
    16. Małgorzata Kobylińska, 2021. "Spatial Diversity of Organic Farming in Poland," Sustainability, MDPI, vol. 13(16), pages 1-19, August.
    17. Kuelbs, James & Zinn, Joel, 2016. "Convergence of quantile and depth regions," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3681-3700.
    18. repec:cte:wsrepe:ws066919 is not listed on IDEAS
    19. 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.
    20. Małgorzata Kobylińska, 2018. "Concept of Observation Depth Measure in the Statistical Analysis of E-Commerce Data in Enterprises," Collegium of Economic Analysis Annals, Warsaw School of Economics, Collegium of Economic Analysis, issue 49, pages 515-526.
    21. Massé, Jean-Claude, 2002. "Asymptotics for the Tukey Median," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 286-300, May.
    22. Cascos, Ignacio & Ochoa, Maicol, 2021. "Expectile depth: Theory and computation for bivariate datasets," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    23. 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.

    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:77:y:2001:i:1:p:138-161. 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.