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From Depth to Local Depth: A Focus on Centrality

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  • Davy Paindaveine
  • Germain Van bever

Abstract

Aiming at analyzing multimodal or nonconvexly supported distributions through data depth, we introduce a local extension of depth. Our construction is obtained by conditioning the distribution to appropriate depth-based neighborhoods and has the advantages, among others, of maintaining affine-invariance and applying to all depths in a generic way. Most importantly, unlike their competitors, which (for extreme localization) rather measure probability mass, the resulting local depths focus on centrality and remain of a genuine depth nature at any locality level. We derive their main properties, establish consistency of their sample versions, and study their behavior under extreme localization. We present two applications of the proposed local depth (for classification and for symmetry testing), and we extend our construction to the regression depth context. Throughout, we illustrate the results on several datasets, both artificial and real, univariate and multivariate. Supplementary materials for this article are available online.

Suggested Citation

  • Davy Paindaveine & Germain Van bever, 2013. "From Depth to Local Depth: A Focus on Centrality," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(503), pages 1105-1119, September.
  • Handle: RePEc:taf:jnlasa:v:108:y:2013:i:503:p:1105-1119
    DOI: 10.1080/01621459.2013.813390
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    File URL: http://hdl.handle.net/10.1080/01621459.2013.813390
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    Cited by:

    1. repec:exl:29stat:v:19:y:2018:i:2:p:331-350 is not listed on IDEAS
    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. Lillo Rodríguez, Rosa Elvira & Laniado Rodas, Henry & Cabana Garceran del Vall, Elisa, 2017. "Multivariate outlier detection based on a robust Mahalanobis distance with shrinkage estimators," DES - Working Papers. Statistics and Econometrics. WS 24613, Universidad Carlos III de Madrid. Departamento de Estadística.
    4. Daniel Kosiorowski & Jerzy P. Rydlewski & Ma{l}gorzata Snarska, 2016. "Detecting a Structural Change in Functional Time Series Using Local Wilcoxon Statistic," Papers 1604.03776, arXiv.org, revised Nov 2016.
    5. Kosiorowska Ewa & Kosiorowski Daniel & Zawadzki Zygmunt, 2015. "Evaluation of the Fourth Millennium Development Goal Realisation using Robust and Nonparametric Tools offered by a Data Depth Concept," Folia Oeconomica Stetinensia, Sciendo, vol. 15(1), pages 34-52, June.
    6. repec:spr:compst:v:32:y:2017:i:3:d:10.1007_s00180-016-0708-9 is not listed on IDEAS
    7. Daniel Kosiorowski, 2015. "Two procedures for robust monitoring of probability distributions of economic data stream induced by depth functions," Operations Research and Decisions, Wroclaw University of Technology, Institute of Organization and Management, vol. 1, pages 55-79.
    8. Davy Paindaveine & Germain Van Bever, 2015. "Discussion of “Multivariate Functional Outlier Detection”, by Mia Hubert, Peter Rousseeuw and Pieter Segaert," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 223-231, July.
    9. Davy Paindaveine & Germain Van Bever, 2017. "Halfspace Depths for Scatter, Concentration and Shape Matrices," Working Papers ECARES ECARES 2017-19, ULB -- Universite Libre de Bruxelles.
    10. repec:bla:istatr:v:85:y:2017:i:1:p:40-43 is not listed on IDEAS
    11. repec:eee:jmvana:v:157:y:2017:i:c:p:53-69 is not listed on IDEAS
    12. Daniel Kosiorowski & Dominik Mielczarek & Jerzy P. Rydlewski, 2017. "Aggregated moving functional median in robust prediction of hierarchical functional time series - an application to forecasting web portal users behaviors," Papers 1710.02669, arXiv.org, revised Jul 2018.
    13. Rainer Dyckerhoff & Christophe Ley & Davy Paindaveine, 2015. "Depth-based runs tests for bivariate central symmetry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(5), pages 917-941, October.

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