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

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.