Aumann and Fréchet Means Overview

• Aumann and Fréchet means are expectation operators for set-valued random variables, defined respectively via Minkowski summation of selections and minimization of expected squared distances.
• Their equivalence under suitable regularity conditions provides a unified framework that ensures unique, robust means and facilitates inference in regression and econometric models.
• By leveraging Hilbert space embeddings and support function metrics, this approach enhances convergence analysis and extends to partial identification in econometric applications.

Aumann and Fréchet means are two fundamental approaches to defining expectation operators for set-valued random variables (SVRVs) arising in probability theory, statistics, and econometrics. The Aumann mean is the classical notion for SVRVs with convex values, grounded in the structure of selections and Minkowski summation, while the Fréchet mean extends the concept of “mean” to more general metric spaces by minimizing expected squared distance. Recent results establish their equivalence under suitable regularity conditions, which provides a unified framework for inference, regression, and identification analysis involving random sets (Kurisu et al., 17 Nov 2025).

1. Definitions: Aumann Mean and Fréchet Mean

Let $(\Omega, \mathcal{A}, \mu)$ denote a probability space and $K_{kc}(\mathbb{R}^d)$ the collection of all nonempty compact convex subsets of $\mathbb{R}^d$. An SVRV is a measurable function $F: \Omega \to K_{kc}(\mathbb{R}^d)$. The selection set is

$$S(F) = \{f: \Omega \to \mathbb{R}^d \text{ measurable}: f(\omega) \in F(\omega) \text{ a.s.} \},\quad S^1(F) = S(F) \cap L^1(\Omega, \mathbb{R}^d).$$

If $F$ is integrably bounded (that is, $\sup_{x \in F(\omega)} \|x\| \in L^1$), the Aumann mean is

$$\mathbb{E}[F] = \{ \mathbb{E}[f] : f \in S^1(F) \} \subset \mathbb{R}^d,$$

which is a convex set. For more general random objects in complete separable metric spaces $(M,d)$, the Fréchet mean is any minimizer

$$\mathbb{E}_+[Y] \in \operatorname*{argmin}_{\nu \in M} \mathbb{E}[d^2(Y, \nu)].$$

The conditional Fréchet mean is analogously defined when conditioning on auxiliary variables.

2. Metric Structure and Support Functions

The support function of $F \in K_{kc}(\mathbb{R}^d)$ is

$$s(p, F) = \sup_{x \in F} \langle p, x \rangle, \quad p \in S^{d-1},$$

with $S^{d-1}$ the unit sphere. The support-function metric between $F, G \in K_{kc}(\mathbb{R}^d)$ is given by

$$d_{kc}(F,G) = \left( \int_{S^{d-1}} [s(p, F) - s(p, G)]^2 dp \right)^{1/2},$$

which metrizes $K_{kc}(\mathbb{R}^d)$. The bounded family $$K_{kc}^B(\mathbb{R}^d) = \{F \in K_{kc}(\mathbb{R}^d): \sup_{x \in F} \|x\| \leq B\}$$ is closed, convex, bounded, and complete under $d_{kc}$.

Convergence in the Hausdorff metric is equivalent, for uniformly bounded sets, to convergence in the $L^2$ sense of support functions. This endows the space of compact convex sets with a robust metric geometry, enabling the use of Hilbert space embeddings in subsequent analysis.

3. Isometric Embedding and Characterization of Fréchet Means

A Hilbert space representation is central: suppose there exists an injective, continuous map $\Psi: M \to H$ into a Hilbert space $(H, \langle \cdot, \cdot \rangle)$ such that

$$d(\alpha, \beta) = \|\Psi(\alpha) - \Psi(\beta)\|_H, \quad \Psi(M) \text{ closed and convex},$$

then for $Y \in M$ with $\mathbb{E}[\|\Psi(Y)\|_H] < \infty$,

$$\Psi(\mathbb{E}_+[Y]) = \mathbb{E}[\Psi(Y)],\quad \mathbb{E}_+[Y] = \Psi^{-1}( \mathbb{E}[\Psi(Y)] ).$$

The law of iterated expectation also holds in the image: $$\Psi(\mathbb{E}_+[\mathbb{E}_+[Y|X]]) = \mathbb{E}[\Psi(Y)]$$.

For random sets, take $M = K_{kc}^B(\mathbb{R}^d)$, $H = L^2(S^{d-1})$, and $\Psi(F) = s(\cdot, F)$. Then $\Psi$ is an isometry onto a closed convex subset of $L^2$.

4. Equivalence of Aumann and Fréchet Means

Via Artstein’s theorem, for integrably bounded SVRV $F$,

$$\mathbb{E}[s(p, F)] = s(p, \mathbb{E}[F]), \quad \forall p,$$

so the mean of support functions equals the support function of the Aumann mean. Consequently,

$$\mathbb{E}_+[F] = \Psi^{-1}(\mathbb{E}[\Psi(F)]) = \mathbb{E}[F].$$

This equivalence holds both unconditionally and conditionally (on a covariate $X$), and ensures uniqueness of both means under integrability and boundedness:

• Both Aumann and Fréchet means (and their conditional analogues) under $d_{kc}$ exist and are unique.
• The law of iterated Fréchet expectation coincides with the Aumann law.

For i.i.d. random sets $F_1, \ldots, F_n$ in $K_{kc}^B(\mathbb{R}^d)$, the unique sample Fréchet mean is

$$\mu_n = \operatorname*{argmin}_\nu \frac{1}{n} \sum_{i=1}^n d_{kc}^2(\nu, F_i) = (1/n) \bigoplus_{i=1}^n F_i,$$

coinciding with the Minkowski average.

5. Explicit Examples and Regression with Random Sets

In the one-dimensional case ($d=1$), every $F \in K_{kc}(\mathbb{R})$ is an interval $[L, U]$, with support function $s(1, F) = U$, $s(-1, F) = -L$. Hence,

$\mathbb{E}[F] = [\mathbb{E}[L], \mathbb{E}[U]],$

which also coincides with the Fréchet mean under $d_{kc}$.

In global Fréchet regression, the criterion is weighted by a function $w(x,X)$, for which

$$m_+(x) = \operatorname*{argmin}_\nu \mathbb{E}[w(x,X) d^2(\nu, F)].$$

Through the embedding, if $w(x, \cdot) \geq 1$,

$$m_+(x) = \mathbb{E}[w(x, X) F] = \mathbb{E}[F] \oplus (x - \mu)^\top \Sigma^{-1} \mathbb{E}[ (X-\mu)F ],$$

aligning with the set-valued best linear predictor of Beresteanu–Molinari (2008). If $w$ may dip below $1$, a Hilbert projection step is required, but the characterization remains explicit.

6. Applications in Econometrics: Partial Identification and Missing Data

In partially identified econometric models, the Aumann mean frequently appears as the identified set for a parameter modeled as a random set; the sample analog is the Minkowski average. By recognizing the equivalence to Fréchet means, metric-statistical techniques become available, such as:

• Global Fréchet regression for set-valued predictors
• Inverse-probability-weighted Fréchet means for “missing at random” SVRV data
• Errors-in-variables and instrumental-variable frameworks via general weighting $w(x,\cdot)$

Convergence rates for these estimators are provided under covering-number assumptions, with near-parametric rates achievable when $d \leq 4$ in low-dimensional settings.

7. Historical Context and Foundational Results

The notion of the Aumann mean was introduced in Aumann (1965), building on the concept of selections for set-valued mappings. Fréchet (1948) proposed the mean as a minimizer of expected squared distance in metric spaces. Artstein (1974) proved the support function expectation result foundational for the equivalence. The set-valued best linear predictor and regression frameworks were established in Beresteanu–Molinari (2008). The unification of these notions via metric statistics and Hilbert embeddings is detailed in Kurisu–Okamoto–Otsu (Kurisu et al., 17 Nov 2025). The framework systematically connects random sets, their metric geometry, and applications in partially identified econometric models.