Bridging Centrality and Extremity: Refining Empirical Data Depth using Extreme Value Statistics

J.H.J. Einmahl, Jun Li, Regina Liu

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Abstract

Abstract: Data depth measures the centrality of a point with respect to a given distribution or data cloud. It provides a natural center-outward ordering of multivariate data points and yields a systematic nonparametric multivariate analysis scheme. In particular, the halfspace depth is shown to have many desirable properties and broad applicability. However, the empirical halfspace depth is zero outside the convex hull of the data. This property has rendered the empirical halfspace depth useless outside the data cloud, and limited its utility in applications where the extreme outlying probability mass is the focal point, such as in classification problems and control charts with very small false alarm rates. To address this issue, we apply extreme value statistics to refine the empirical halfspace depth in “the tail”. This provides an important linkage between data depth, which is useful for inference on centrality, and extreme value statistics, which is useful for inference on extremity. The refined empirical halfspace depth can thus extend all its utilities beyond the data cloud, and hence broaden greatly its applicability. The refined estimator is shown to have substantially improved upon the empirical estimator in theory and simulations. The benefit of this improvement is also demonstrated through the applications in classification and statistical process control.
Original languageEnglish
Place of PublicationTilburg
PublisherEconometrics
Number of pages30
Volume2015-020
Publication statusPublished - 23 Mar 2015

Publication series

NameCentER Discussion Paper
Volume2015-020

Fingerprint

hull
statistics
refining
multivariate analysis
simulation
distribution
rate
alarm

Keywords

  • depth
  • extremes
  • nonparametric classification
  • nonparametric multivariate SPC
  • tail

Cite this

Einmahl, J. H. J., Li, J., & Liu, R. (2015). Bridging Centrality and Extremity: Refining Empirical Data Depth using Extreme Value Statistics. (CentER Discussion Paper; Vol. 2015-020). Tilburg: Econometrics.
Einmahl, J.H.J. ; Li, Jun ; Liu, Regina. / Bridging Centrality and Extremity : Refining Empirical Data Depth using Extreme Value Statistics. Tilburg : Econometrics, 2015. (CentER Discussion Paper).
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abstract = "Abstract: Data depth measures the centrality of a point with respect to a given distribution or data cloud. It provides a natural center-outward ordering of multivariate data points and yields a systematic nonparametric multivariate analysis scheme. In particular, the halfspace depth is shown to have many desirable properties and broad applicability. However, the empirical halfspace depth is zero outside the convex hull of the data. This property has rendered the empirical halfspace depth useless outside the data cloud, and limited its utility in applications where the extreme outlying probability mass is the focal point, such as in classification problems and control charts with very small false alarm rates. To address this issue, we apply extreme value statistics to refine the empirical halfspace depth in “the tail”. This provides an important linkage between data depth, which is useful for inference on centrality, and extreme value statistics, which is useful for inference on extremity. The refined empirical halfspace depth can thus extend all its utilities beyond the data cloud, and hence broaden greatly its applicability. The refined estimator is shown to have substantially improved upon the empirical estimator in theory and simulations. The benefit of this improvement is also demonstrated through the applications in classification and statistical process control.",
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Einmahl, JHJ, Li, J & Liu, R 2015 'Bridging Centrality and Extremity: Refining Empirical Data Depth using Extreme Value Statistics' CentER Discussion Paper, vol. 2015-020, Econometrics, Tilburg.

Bridging Centrality and Extremity : Refining Empirical Data Depth using Extreme Value Statistics. / Einmahl, J.H.J.; Li, Jun; Liu, Regina.

Tilburg : Econometrics, 2015. (CentER Discussion Paper; Vol. 2015-020).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

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AU - Liu, Regina

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N2 - Abstract: Data depth measures the centrality of a point with respect to a given distribution or data cloud. It provides a natural center-outward ordering of multivariate data points and yields a systematic nonparametric multivariate analysis scheme. In particular, the halfspace depth is shown to have many desirable properties and broad applicability. However, the empirical halfspace depth is zero outside the convex hull of the data. This property has rendered the empirical halfspace depth useless outside the data cloud, and limited its utility in applications where the extreme outlying probability mass is the focal point, such as in classification problems and control charts with very small false alarm rates. To address this issue, we apply extreme value statistics to refine the empirical halfspace depth in “the tail”. This provides an important linkage between data depth, which is useful for inference on centrality, and extreme value statistics, which is useful for inference on extremity. The refined empirical halfspace depth can thus extend all its utilities beyond the data cloud, and hence broaden greatly its applicability. The refined estimator is shown to have substantially improved upon the empirical estimator in theory and simulations. The benefit of this improvement is also demonstrated through the applications in classification and statistical process control.

AB - Abstract: Data depth measures the centrality of a point with respect to a given distribution or data cloud. It provides a natural center-outward ordering of multivariate data points and yields a systematic nonparametric multivariate analysis scheme. In particular, the halfspace depth is shown to have many desirable properties and broad applicability. However, the empirical halfspace depth is zero outside the convex hull of the data. This property has rendered the empirical halfspace depth useless outside the data cloud, and limited its utility in applications where the extreme outlying probability mass is the focal point, such as in classification problems and control charts with very small false alarm rates. To address this issue, we apply extreme value statistics to refine the empirical halfspace depth in “the tail”. This provides an important linkage between data depth, which is useful for inference on centrality, and extreme value statistics, which is useful for inference on extremity. The refined empirical halfspace depth can thus extend all its utilities beyond the data cloud, and hence broaden greatly its applicability. The refined estimator is shown to have substantially improved upon the empirical estimator in theory and simulations. The benefit of this improvement is also demonstrated through the applications in classification and statistical process control.

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Einmahl JHJ, Li J, Liu R. Bridging Centrality and Extremity: Refining Empirical Data Depth using Extreme Value Statistics. Tilburg: Econometrics. 2015 Mar 23. (CentER Discussion Paper).