  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.
