IDEAS home Printed from https://ideas.repec.org/p/zbw/sfb475/200257.html
   My bibliography  Save this paper

Breakdown and groups

Author

Listed:
  • Davies, P. Laurie
  • Gather, U.

Abstract

The concept of breakdown point was introduced by Hodges (1967) and Hampel (1968, 1971) and still plays an important though at times a controversial role in robust statistics. It has proved most successful in the context of location, scale and regression problems. In this paper we argue that this success is intimately connected to the fact that the translation and affine groups act on the sample space and give rise to a definition of equivariance for statistical functionals. For such functionals a nontrivial upper bound for the breakdown point can be shown. In the absence of such a group structure a breakdown point of one is attainable and this is perhaps the decisive reason why the concept of breakdown point in other situations has not proved as successful. Even if a natural group is present it is often not sufficiently large to allow a nontrivial upper bound for the breakdown point. One exception to this is the problem of the autocorrelation structure of time series where we derive a nontrivial upper breakdown point using the group of realizable linear filters. The paper is formulated in an abstract manner to emphasize the role of the group and the resulting equivariance structure.

Suggested Citation

  • Davies, P. Laurie & Gather, U., 2002. "Breakdown and groups," Technical Reports 2002,57, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    • Handle: RePEc:zbw:sfb475:200257
    as

    Download full text from publisher

    File URL: https://www.econstor.eu/bitstream/10419/77370/2/2002-57.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Davies, P. Laurie & Kovac, A., 2002. "Densities, spectral densities and modality," Technical Reports 2002,53, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    2. He, Xuming & Fung, Wing K., 2000. "High Breakdown Estimation for Multiple Populations with Applications to Discriminant Analysis," Journal of Multivariate Analysis, Elsevier, vol. 72(2), pages 151-162, February.
    3. Hubert, Mia, 1997. "The breakdown value of the L1 estimator in contingency tables," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 419-425, May.
    4. Gordaliza, A., 1991. "On the breakdown point of multivariate location estimators based on trimming procedures," Statistics & Probability Letters, Elsevier, vol. 11(5), pages 387-394, May.
    5. Yanyuan Ma & Marc G. Genton, 2000. "Highly Robust Estimation of the Autocovariance Function," Journal of Time Series Analysis, Wiley Blackwell, vol. 21(6), pages 663-684, November.
    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. Davies, P. Laurie, 2002. "Statistical procedures and robust statistics," Technical Reports 2002,54, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.

    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. Trucíos, Carlos & Mazzeu, João H.G. & Hotta, Luiz K. & Valls Pereira, Pedro L. & Hallin, Marc, 2021. "Robustness and the general dynamic factor model with infinite-dimensional space: Identification, estimation, and forecasting," International Journal of Forecasting, Elsevier, vol. 37(4), pages 1520-1534.
    2. Tino Werner, 2023. "Quantitative robustness of instance ranking problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(2), pages 335-368, April.
    3. Huang, Yufen & Cheng, Ching-Ren & Wang, Tai-Ho, 2008. "Pair-perturbation influence functions of nongaussianity by projection pursuit," Computational Statistics & Data Analysis, Elsevier, vol. 52(8), pages 3971-3987, April.
    4. Ollinger, Michael, 2024. "Recall characteristics and food safety process control," Food Policy, Elsevier, vol. 124(C).
    5. Trucíos, Carlos & Hotta, Luiz K. & Valls Pereira, Pedro L., 2019. "On the robustness of the principal volatility components," Journal of Empirical Finance, Elsevier, vol. 52(C), pages 201-219.
    6. Carlos Cesar Trucios-Maza & João H. G Mazzeu & Luis K. Hotta & Pedro L. Valls Pereira & Marc Hallin, 2019. "On the robustness of the general dynamic factor model with infinite-dimensional space: identification, estimation, and forecasting," Working Papers ECARES 2019-32, ULB -- Universite Libre de Bruxelles.
    7. Eichler, M. & Türk, D., 2013. "Fitting semiparametric Markov regime-switching models to electricity spot prices," Energy Economics, Elsevier, vol. 36(C), pages 614-624.
    8. Olcay Arslan, 2010. "An alternative multivariate skew Laplace distribution: properties and estimation," Statistical Papers, Springer, vol. 51(4), pages 865-887, December.
    9. Zuo, Yijun, 2001. "Some quantitative relationships between two types of finite sample breakdown point," Statistics & Probability Letters, Elsevier, vol. 51(4), pages 369-375, February.
    10. Reisen, Valdério Anselmo & Sgrancio, Adriano Marcio & Lévy-Leduc, Céline & Bondon, Pascal & Monte, Edson Zambon & Aranda Cotta, Higor Henrique & Ziegelmann, Flávio Augusto, 2019. "Robust factor modelling for high-dimensional time series: An application to air pollution data," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 842-852.
    11. Fajardo, Fabio Alexander, 2011. "Some Alternatives for Robust Estimation of the Spectrum in Stationary Processes," Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, vol. 31(1), March.
    12. Boudt, Kris & Heyndels, Ewoud, 2024. "Robust interactive fixed effects," Econometrics and Statistics, Elsevier, vol. 29(C), pages 206-223.
    13. Cristian F. Jiménez‐Varón & Fouzi Harrou & Ying Sun, 2024. "Pointwise data depth for univariate and multivariate functional outlier detection," Environmetrics, John Wiley & Sons, Ltd., vol. 35(5), August.
    14. Todorov, Valentin & Filzmoser, Peter, 2010. "Robust statistic for the one-way MANOVA," Computational Statistics & Data Analysis, Elsevier, vol. 54(1), pages 37-48, January.
    15. Mia Hubert & Stephan Van der Veeken, 2010. "Robust classification for skewed data," 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. 4(4), pages 239-254, December.
    16. Stephen Machin & Olivier Marie & Sunčica Vujić, 2012. "Youth Crime and Education Expansion," German Economic Review, Verein für Socialpolitik, vol. 13(4), pages 366-384, November.
    17. Bashir, Shaheena & Carter, E. M., 2005. "High breakdown mixture discriminant analysis," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 102-111, March.
    18. Fried, Roland & Gather, Ursula, 2004. "Robust Trend Estimation for AR(1) Disturbances," Technical Reports 2004,64, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    19. Pires, Ana M. & Branco, João A., 2002. "Partial Influence Functions," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 451-468, November.
    20. Matías Salibián-Barrera & Stefan Aelst & Gert Willems, 2008. "Fast and robust bootstrap," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 17(1), pages 41-71, February.

    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:zbw:sfb475:200257. 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: ZBW - Leibniz Information Centre for Economics (email available below). General contact details of provider: https://edirc.repec.org/data/isdorde.html .

    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.