A Note on Restricted Selection Set from Random Interval

TL;DR

This note develops a sharp, one-dimensional analysis of restricted selection sets for interval-valued random sets on a non-atomic space. It shows that the classical Aumann expectation and median bounds are sharp and that functional values such as means and quantiles can be attained under mean or median constraints, with explicit formulas and dual representations for event probabilities. The authors derive threshold-calibration rules and quantile-integral expressions to characterize how restrictions shrink or preserve identifiability, and they extend the framework to higher moments and non-median quantiles. Overall, the work clarifies identifiability and extremal behavior of latent variables constrained to random intervals under partial information, with potential applications in economics and statistics.

Abstract

We study restricted selection sets of random intervals in $\R^1$ defined on a non-atomic probability space. Given a random interval $Y=[y_L,y_U]$ and scalar constraints on the expectation or the median of admissible selections, we define the restricted selection set and establish its existence, basic structure and influence on bounding moments and quantiles. In particular, we give conditions under which any mean (or quantile) in the Aumann expectation range can be attained by a measurable selection. We characterize the induced ranges of means, medians, and event probabilities. The analysis is carried out in a minimal one-dimensional random-set framework inspired by the classical theory of Aumann integrals. We also outline extensions to higher-order moment and general quantile restrictions.

Figures (1)

• Figure 1: Schematic construction of the extremal boundary distributions in Example \ref{['ex:schematic']}. Thin black and gray curves show the baseline CDFs $F_L$ and $F_U$ of the chi-square lower and upper bounds. The bold green (resp. red) segments indicate the local modification of $F_L$ on $[m_L,m]$ (resp. of $F_U$ on $[m,m_U]$) required to satisfy the median constraints $\mathbf{P}(y\le m)=\mathbf{P}(y\ge m)=\tfrac12$ for an underlying selection $y\in\mathop{\mathrm{\mathbf{Sel}}}\nolimits(Y)$. Vertical lines at $m_L$, $m$, and $m_U$ and the bullets at height $1/2$ highlight the binding median restrictions.

Theorems & Definitions (42)

• Definition 2.1: Mean-restricted selection set
• Proposition 2.2
• proof
• Definition 2.3: Median-restricted selection set
• Proposition 2.4: Quantile attainability via selections
• proof
• Lemma 2.5: Conditional rearrangement / quantile boundLiebLoss2001
• proof
• Lemma 2.6: Quantile--area identity HardyLittlewoodPolya1952
• proof