Properties of statistical depth with respect to compact convex random sets. The Tukey depth

TL;DR

This paper introduces a statistical data depth for compact convex random sets, extending the multivariate Tukey depth and fuzzy set depth, and establishes its fundamental properties.

Contribution

It provides a new depth function for compact convex random sets, aligning with existing depth notions and expanding their theoretical framework.

Findings

The depth is consistent with multivariate Tukey depth.

The depth satisfies key axiomatic properties.

The paper extends depth concepts to convex random sets.

Abstract

We study a statistical data depth with respect to compact convex random sets which is consistent with the multivariate Tukey depth and the Tukey depth for fuzzy sets. In doing so, we provide a series of properties for statistical data depth with respect to compact convex random sets. These properties are an adaptation of properties that constitute the axiomatic notions of multivariate, functional and fuzzy depth functions and other well-known properties of depth.