Aumann Expectation Range in Interval Data

• Aumann Expectation Range is the set of all possible expected values derived from measurable selections of a random interval on a non-atomic probability space.
• It is constructed via convex mixing of the interval endpoints, ensuring that every mean between E(yL) and E(yU) is attained.
• This concept underpins partial identification approaches in econometrics by providing precise bounds for latent means, quantiles, and higher moments.

The Aumann expectation range describes the set of possible expected values (means) that can be realized by measurable selections from a one-dimensional random interval-valued map on a non-atomic probability space. More formally, if $Y = [y_L, y_U]$ is a random interval defined on $(\Omega, \mathcal{A}, P)$, with $y_L$ and $y_U$ integrable and $y_L(\omega) \leq y_U(\omega)$ almost surely, the Aumann expectation is the closed interval $[E(y_L), E(y_U)]$. Each point in this interval corresponds to the expectation of some measurable selection $y: \Omega \to \mathbb{R}$ with $y_L(\omega) \leq y(\omega) \leq y_U(\omega)$ almost surely, and the set of such selections is closed under convex mixing. The structure and attainability of the Aumann expectation range are central to partial identification analysis and have implications for bounding moments and quantiles in econometrics and statistics (Beresteanu et al., 4 Dec 2025).

1. Formal Definition and Measurable Selections

Let $Y: \Omega \to \mathcal{K}(\mathbb{R})$ be a random set-valued map, where $\mathcal{K}(\mathbb{R})$ denotes the family of nonempty closed subsets of $\mathbb{R}$. A measurable selection is a measurable function $y: \Omega \to \mathbb{R}$ such that $y(\omega) \in Y(\omega)$ almost surely, with $E|y| < \infty$. The set of such selections is denoted

$\Sel^1(Y) = \{ y : \Omega \to \mathbb{R} \mid y \text{ measurable},\, y(\omega) \in Y(\omega)\ \text{a.s.},\, E|y| < \infty \}.$

The Aumann integral of $Y$ is defined as

$\int Y(\omega) dP(\omega) := \left\{ \int y(\omega) dP(\omega) : y \in \Sel^1(Y) \right\}$

and the Aumann expectation is its topological closure in $\mathbb{R}$: $E[Y] := \overline{ \left\{ E[y] : y \in \Sel^1(Y) \right\} }.$ For one-dimensional intervals $Y = [y_L, y_U]$, this closure yields an explicit interval.

2. The One-Dimensional Interval Case

For a random interval $Y = [y_L, y_U]$ on a non-atomic probability space, the Aumann expectation is explicitly

$E[Y] = [E(y_L), E(y_U)].$

This interval is achieved as follows: for any $t \in [0, 1]$, the convex combination $y_t = (1 - t) y_L + t y_U$ defines a measurable selection in $\Sel^1(Y)$ whose expectation is $(1-t)E(y_L) + tE(y_U)$, traversing the interval $[E(y_L), E(y_U)]$ as $t$ moves from $0$ to $1$. The result is based on classical work by Aumann (1965), Artstein (1974), and Molchanov (2005), with a direct proof for the interval case relying only on linearity and convexity (Beresteanu et al., 4 Dec 2025).

3. Attainment of Means and the Structure of Selections

Every $\kappa \in [E(y_L), E(y_U)]$ can be realized as the mean of a measurable selection constructed by interpolating between $y_L$ and $y_U$. Specifically, for $\kappa \in [E(y_L), E(y_U)]$, define

$$t^* = \begin{cases} \frac{\kappa - E(y_L)}{E(y_U) - E(y_L)} & \text{if } E(y_U) \neq E(y_L) \ 0 & \text{if } E(y_U) = E(y_L) \end{cases} \quad y(\omega) = (1 - t^*) y_L(\omega) + t^* y_U(\omega).$$

Then $y \in \Sel^1(Y)$ and $E[y] = \kappa$. This construction guarantees that $\Sel(Y | \kappa) \neq \emptyset$ for every $\kappa$ in the Aumann expectation interval, ensuring that the full range is indeed realized by simple measurable selections (Beresteanu et al., 4 Dec 2025).

4. Examples: Constant and Stochastic Intervals

For a constant interval $Y = [a, b]$ almost surely, every measurable selection is a constant $y \equiv m$ with $m \in [a, b]$, and the Aumann expectation is precisely $[a, b]$. If the endpoints are stochastic, such as

$$y_L = F_{\chi^2_2}^{-1}(U), \quad y_U = F_{\chi^2_5}^{-1}(U), \quad U \sim \mathrm{Uniform}(0, 1),$$

then $E(y_L) \approx 2$, $E(y_U) \approx 5$, and $E[Y] = [2, 5]$. As $t$ varies from $0$ to $1$, the interpolated selection $y_t = (1 - t)y_L + t y_U$ smoothly traces the mean range $[2, 5]$ (Beresteanu et al., 4 Dec 2025).

5. Extensions: Higher Moments and Quantile Constraints

The analysis generalizes to constraints beyond the mean. If one imposes a restriction on a higher moment,

$E[y^r] = \mu_r \text{ for } r > 0,$

then, under monotonicity conditions on $x \mapsto x^r$, the set of attainable expectations is

$\left\{ E[y]: y \in \Sel_r(Y \mid \mu_r) \right\} = \left[ \underline{m}(r, \mu_r),\ \overline{m}(r, \mu_r) \right],$

with $\overline{m}(r, \mu_r)$ characterized as

$$\overline{m}(r, \mu_r) = \inf_{\lambda \in \mathbb{R}} \left\{ E\left[ \sup_{x \in Y(\omega)} (x + \lambda x^r) \right] - \lambda \mu_r \right\}.$$

Similarly, fixing an $\alpha$-quantile $q_\alpha$ within $[T_Y^{-1}(\alpha), C_Y^{-1}(\alpha)]$ leads to a nonempty selection set $\Sel_\alpha(Y \mid q_\alpha)$ and a mean range that is again a convex compact subinterval of $[E(y_L), E(y_U)]$. Extremal selections for quantile constraints are constructed by appropriate rearrangements over upper and lower tails (Beresteanu et al., 4 Dec 2025).

6. Significance and Applications

The reduction of the Aumann expectation for a random interval to the explicit interval $[E(y_L), E(y_U)]$—with full attainability of means in the presence of a non-atomic probability space—underpins partial identification methodology in economics and statistics. When only interval-valued data are available, the structure of the Aumann expectation range guarantees that all possible population means and quantiles consistent with the partial observability can be operationally realized as expectations of measurable selections. This property facilitates sharp bounding arguments for latent variables and quantifies the informational content in models with incomplete data (Beresteanu et al., 4 Dec 2025).