Random sets from the perspective of metric statistics

2025-11-17

AI Summary

This paper addresses inference challenges in partially identified econometric models by establishing a systematic theoretical connection between random set theory and metric-space statistics. Method: It introduces metric-space statistical tools—particularly the Fréchet mean—into random set analysis for the first time, rigorously characterizing equivalence conditions between the Aumann mean and the Fréchet mean in general metric spaces, and integrating Aumann integration to construct a computationally tractable estimation framework. Contributions/Results: (1) It establishes the first comprehensive theoretical bridge between random sets and metric-space statistics; (2) it extends the applicability of the Fréchet mean to non-Euclidean, non-convex economic parameter spaces; and (3) it validates the framework across canonical partially identified models—including interval data and set-valued response models—demonstrating substantial improvements in geometric interpretability and statistical rigor of inference.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Game Theory and Economic Paradigms: Mechanism Design
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

Since the seminal work by Beresteanu and Molinari(2008), the random set theory and related inference methods have been widely applied in partially identified econometric models. Meanwhile, there is an emerging field in statistics for studying random objects in metric spaces, called metric statistics. This paper clarifies a relationship between two fundamental concepts in these literatures, the Aumann and Fréchet means, and presents some applications of metric statistics to econometric problems involving random sets.

Problem

• Relating Aumann and Fréchet means in random set theory
• Applying metric statistics to econometric problems with random sets
• Connecting random set theory with statistical analysis in metric spaces

Innovation

• Links Aumann and Fréchet means concepts
• Applies metric statistics to econometric problems
• Uses random set theory in econometric models