Lift expectations of random sets

TL;DR

This paper generalizes the concept of lift zonoids to random convex bodies via lift expectations, revealing limitations in uniquely identifying distributions and exploring implications for statistical depth regions and orderings.

Contribution

It introduces the lift expectation for random convex bodies and analyzes its non-uniqueness in identifying distributions, extending the theory beyond random vectors.

Findings

Lift expectation uniquely determines one-dimensional support functions.

Different convex bodies can share the same lift expectation.

Applications include depth-trimmed regions and partial orderings.

Abstract

It is known that the distribution of an integrable random vector $\xi$ in $\mathbb{R}^d$ is uniquely determined by a $(d+1)$-dimensional convex body called the lift zonoid of $\xi$. This concept is generalised to define the lift expectation of random convex bodies. However, the unique identification property of distributions is lost; it is shown that the lift expectation uniquely identifies only one-dimensional distributions of the support function, and so different random convex bodies may share the same lift expectation. The extent of this nonuniqueness is analysed and it is related to the identification of random convex functions using only their one-dimensional marginals. Applications to construction of depth-trimmed regions and partial ordering of random convex bodies are also mentioned.