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Lens depth, a measure in metric spaces, is now defined for general spaces. Its empirical form converges to the true lens depth, with level sets consistently estimating population counterparts.

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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.