Convergence of depth contours for multivariate datasets

Abstract

Contours of depth often provide a good geometrical understanding of the structure of a multivariate dataset. They are also useful in robust statistics in connection with generalized medians and data ordering. If the data constitute a random sample from a spherical or elliptic distribution, the depth contours are generally required to converge to spherical or elliptical shapes. We consider contour constructions based on a notion of data depth and prove a uniform contour convergence theorem under verifiable conditions on the depth measure. Applications to several existing depth measures discussed in the literature are also considered.