The Annals of Statistics

Bridging centrality and extremity: Refining empirical data depth using extreme value statistics

John H. J. Einmahl, Jun Li, and Regina Y. Liu

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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.

Article information

Ann. Statist., Volume 43, Number 6 (2015), 2738-2765.

Received: September 2014
Revised: June 2015
First available in Project Euclid: 7 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties 62G32: Statistics of extreme values; tail inference
Secondary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 62P30: Applications in engineering and industry

Depth extremes nonparametric classification nonparametric multivariate SPC tail


Einmahl, John H. J.; Li, Jun; Liu, Regina Y. Bridging centrality and extremity: Refining empirical data depth using extreme value statistics. Ann. Statist. 43 (2015), no. 6, 2738--2765. doi:10.1214/15-AOS1359.

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