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Random sets from the perspective of metric statistics

Daisuke Kurisu, Yuta Okamoto, Taisuke Otsu

TL;DR

The paper unifies random-set analysis with metric-statistics by proving that the Aumann mean and Fréchet mean coincide for random compact convex sets under the $L^2$ support-function metric, via an isometric embedding into a Hilbert space. It then extends global Fréchet regression to set-valued outcomes, showing how $m_\oplus(x)=\\Psi^{-1}(\\mathbb{E}[w(x,X)\\Psi(F)])$ links to the population best linear predictor through Minkowski addition, with explicit forms when the weight satisfies $w(x,X)\ge 1$. The work also develops extensions to practical econometric problems, including errors-in-variables and missing data, using adjusted weights and inverse-probability weighting to maintain consistency and derive rates of convergence for the Fréchet-mean estimators. Taken together, these results provide a tractable, theory-backed framework for conducting regression and inference on random set-valued data within econometric and statistical applications. The methods enable principled analysis of partially identified models and complex set-valued outcomes in economics and related fields.

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.

Random sets from the perspective of metric statistics

TL;DR

The paper unifies random-set analysis with metric-statistics by proving that the Aumann mean and Fréchet mean coincide for random compact convex sets under the support-function metric, via an isometric embedding into a Hilbert space. It then extends global Fréchet regression to set-valued outcomes, showing how links to the population best linear predictor through Minkowski addition, with explicit forms when the weight satisfies . The work also develops extensions to practical econometric problems, including errors-in-variables and missing data, using adjusted weights and inverse-probability weighting to maintain consistency and derive rates of convergence for the Fréchet-mean estimators. Taken together, these results provide a tractable, theory-backed framework for conducting regression and inference on random set-valued data within econometric and statistical applications. The methods enable principled analysis of partially identified models and complex set-valued outcomes in economics and related fields.

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.

Paper Structure

This paper contains 33 sections, 17 theorems, 52 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Setup and Aumann mean
  4. Support function and metric on $K_{kc}(\mathbb{R}^d)$
  5. Fréchet mean and isometric embedding
  6. Equivalence of Aumann and Fréchet means
  7. Metric statistics for random sets
  8. Linear regression as a weighted Fréchet mean
  9. Global Fréchet regression for random sets
  10. Illustration
  11. Errors-in-variables
  12. Missing data
  13. Proofs
  14. Proof of Lemma \ref{['lem:d']}
  15. Proof of (i)
  16. ...and 18 more sections

Key Result

Lemma 2.1

$d_{kc}$ is a metric on $K_{kc}(\mathbb{R}^d)$. Let $\{F_n\}_{n \geq 0} \subset K_{kc}^B(\mathbb{R}^d)$ be a sequence of compact convex sets. Then $\lim_{n \to \infty}d_{kc}(F_n, F_0) = 0$ if and only if $\lim_{n \to \infty}d_H(F_n, F_0) = 0$. $K_{kc}^B(\mathbb{R}^d)$ is a bounded closed convex subs

Figures (2)

  • Figure 3.1: Illustration of the relation between $m_\oplus(x)$ and $\mathbb{E}[w(x,X)F]$ when $d=1$.
  • Figure 3.2: Weight functions $w(\mu,z)$ and $w(16,z)$

Theorems & Definitions (27)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.1
  • Corollary 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma B.1
  • proof
  • Lemma B.2
  • ...and 17 more