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

TL;DR

This paper enhances the empirical half-space depth measure by integrating extreme value statistics, enabling better analysis of outliers and extremities beyond the data cloud for applications like classification and control charts.

Contribution

It introduces a novel method that refines empirical data depth using extreme value theory, extending its utility to the tails and outliers.

Findings

Refined depth estimator outperforms the empirical version in theory and simulations.

Improved method enhances outlier detection in classification and control charts.

Extension of data depth utility beyond the data cloud broadens application scope.

Abstract

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.