Open Access
December 2015 Bridging centrality and extremity: Refining empirical data depth using extreme value statistics
John H. J. Einmahl, Jun Li, Regina Y. Liu
Ann. Statist. 43(6): 2738-2765 (December 2015). DOI: 10.1214/15-AOS1359


Statistical 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 half-space depth is shown to have many desirable properties and broad applicability. However, the empirical half-space depth is zero outside the convex hull of the data. This property has rendered the empirical half-space 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 half-space 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 half-space 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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John H. J. Einmahl. Jun Li. Regina Y. Liu. "Bridging centrality and extremity: Refining empirical data depth using extreme value statistics." Ann. Statist. 43 (6) 2738 - 2765, December 2015.


Received: 1 September 2014; Revised: 1 June 2015; Published: December 2015
First available in Project Euclid: 7 October 2015

zbMATH: 1327.62205
MathSciNet: MR3405610
Digital Object Identifier: 10.1214/15-AOS1359

Primary: 62G05 , 62G20 , 62G32
Secondary: 62H30 , 62P30

Keywords: depth , Extremes , Nonparametric classification , nonparametric multivariate SPC , tail

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 6 • December 2015
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