Level sets of depth measures in abstract spaces

TL;DR

This paper investigates the properties of level sets of the lens depth in abstract metric spaces, demonstrating their consistency and boundary estimation accuracy based on iid samples, with applications to real-world data.

Contribution

It extends the concept of lens depth to general metric spaces and proves the consistency of empirical level sets and their boundaries.

Findings

Empirical level sets are consistent estimators of population level sets.

Boundaries of empirical level sets reliably estimate true boundaries.

Theoretical results are supported by real-life examples.

Abstract

The lens depth of a point has been recently extended to general metric spaces, which is not the case for most depths. It is defined as the probability of being included in the intersection of two random balls centred at two random points X and Y, with the same radius d(X, Y). We study the consistency in Hausdorff and measure distance, of the level sets of the empirical lens depth, based on an iid sample on a general metric space. We also prove that the boundary of the empirical level sets are consistent estimators of their population counterparts, and analyze two real-life examples