Level sets of depth measures in abstract spaces.

Area of Science:

• * Mathematical Statistics
• * Metric Space Theory

Background:

• * Most depth measures are restricted to Euclidean spaces.
• * Lens depth has been extended to general metric spaces.
• * This extension allows for broader applications of depth measures.

Purpose of the Study:

• * To establish theoretical properties of empirical lens depth in general metric spaces.
• * To prove the convergence of empirical lens depth level sets to their population counterparts.
• * To demonstrate the consistency of empirical level sets as estimators for population level sets.

Main Methods:

• * Utilizing the definition of lens depth as the probability of a point being in the intersection of two random balls.
• * Applying convergence theorems in the Painlevé-Kuratowski sense for set convergence.
• * Employing Hausdorff distance to assess the consistency of empirical estimators.

Main Results:

• * Proven convergence of empirical lens depth level sets to population level sets in separable and complete metric spaces.
• * Demonstrated that empirical level sets and their boundaries are consistent estimators in Hausdorff distance for compact sets.
• * Analyzed the practical utility of lens depth through two real-life examples.

Conclusions:

• * The empirical lens depth provides a reliable estimation of the population lens depth in general metric spaces.
• * The theoretical findings support the use of lens depth in statistical analysis within metric spaces.
• * The study validates lens depth as a robust statistical tool with practical applications.