On similarity of the sample depth contours

TL;DR

This paper explores the shape similarity of sample projection depth contours, revealing their proportional relationships and potential for computational efficiency and improved depth measures outside data convex hulls.

Contribution

It demonstrates the shape similarity of projection depth contours and proposes extensions to halfspace and zonoid depths to address the outside problem.

Findings

Contours are similar in shape with different sizes.

Potential for more efficient computation of depth contours.

Extensions to existing depth measures to handle outside the convex hull.

Abstract

In this paper, we investigate the similarity property of the sample projection depth contours. It turns out that some of these contours are of \emph{the same shape} with different sizes, following a similar fashion to the Mahalanobis depth contours. One advantage of this investigation is the potential of bringing convenience to the computation of the projection depth contours; the other one is that we may utilize this idea to extend both the halfspace depth and zonoid depth to versions that do not vanish outside the convex hull of the data cloud, aiming at overcoming the so-called `outside problem'. Examples are also provided to illustrate the main results.