IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v42y1992i1p141-161.html
   My bibliography  Save this article

Bias-robust estimators of multivariate scatter based on projections

Author

Listed:
  • Maronna, Ricardo A.
  • Stahel, Werner A.
  • Yohai, Victor J.

Abstract

Equivariant estimation of the multivariate scatter of a random vector X can be derived from a criterion of (lack of) spherical symmetry g(X). The scatter matrix is V = (ATA)-1, where A is the transformation matrix which makes AX as spherical as possible, that is, which minimizes g(AX). The new class of projection estimators is based on making the spread of univariate projections as constant as possible by choosing g(X) = supu = 1 s(uTX) -1, where s is any robust scale functional. The breakdown point of such an estimator is at least that of s, independently of the dimension p of X. In order to study the bias, we calculate condition numbers based on asymptotics and on simulations of finite samples for a spherically symmetric X, contaminated by a point mass, with the median absolute deviation as the scale measure. The simulations are done for an algorithm which is designed to approximate the projection estimator. The bias is much lower than the one of Rousseeuw's MVE-estimator, and compares favorably in most cases with two M-estimators.

Suggested Citation

  • Maronna, Ricardo A. & Stahel, Werner A. & Yohai, Victor J., 1992. "Bias-robust estimators of multivariate scatter based on projections," Journal of Multivariate Analysis, Elsevier, vol. 42(1), pages 141-161, July.
    • Handle: RePEc:eee:jmvana:v:42:y:1992:i:1:p:141-161
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0047-259X(92)90084-S
    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

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


    Cited by:

    1. Maronna, Ricardo A. & Yohai, Víctor J., 1994. "Robust estimation in simultaneous equations models," DES - Working Papers. Statistics and Econometrics. WS 3956, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. Ma, Yanyuan & Genton, Marc G., 2001. "Highly Robust Estimation of Dispersion Matrices," Journal of Multivariate Analysis, Elsevier, vol. 78(1), pages 11-36, July.
    3. Fekri, M. & Ruiz-Gazen, A., 2004. "Robust weighted orthogonal regression in the errors-in-variables model," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 89-108, January.
    4. Zhang, Jian, 2002. "Some Extensions of Tukey's Depth Function," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 134-165, July.
    5. Tyler, David E., 2010. "A note on multivariate location and scatter statistics for sparse data sets," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1409-1413, September.
    6. Hernández, Sonia & Yohai, Víctor J., 1999. "Locally and globally robust estimators in regression," DES - Working Papers. Statistics and Econometrics. WS 6348, Universidad Carlos III de Madrid. Departamento de Estadística.
    7. Schmitt, Eric & Öllerer, Viktoria & Vakili, Kaveh, 2014. "The finite sample breakdown point of PCS," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 214-220.
    8. Gather, Ursula & Davies, P. Laurie, 2004. "Robust Statistics," Papers 2004,20, Humboldt University of Berlin, Center for Applied Statistics and Economics (CASE).
    9. Croux, Christophe & Haesbroeck, Gentiane, 1997. "An easy way to increase the finite-sample efficiency of the resampled minimum volume ellipsoid estimator," Computational Statistics & Data Analysis, Elsevier, vol. 25(2), pages 125-141, July.
    10. Zhou, Weihua & Dang, Xin, 2010. "Projection based scatter depth functions and associated scatter estimators," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 138-153, January.
    11. David E. Tyler & Frank Critchley & Lutz Dümbgen & Hannu Oja, 2009. "Invariant co‐ordinate selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 549-592, June.
    12. Juan, Jesús & Prieto, Francisco J., 1994. "A subsampling method for the computation of multivariate estimators with high breakdown point," DES - Working Papers. Statistics and Econometrics. WS 3952, Universidad Carlos III de Madrid. Departamento de Estadística.

    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:42:y:1992:i:1:p:141-161. 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.

    We have no bibliographic 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.

    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.