Lee Bounds for Random Objects

2026-01-14

AI Summary

This study extends Lee’s (2009) bounds approach for treatment effects to settings with selective nonresponse and outcomes taking values in general metric spaces—such as compositional or distributional data. By embedding the Fréchet means of potential outcomes into Euclidean or Hilbert spaces, the authors characterize the identified set for the average treatment effect and construct corresponding estimators along with bootstrap-based confidence regions. This work represents the first extension of Lee’s bounds to non-Euclidean random objects, thereby overcoming the conventional restriction to real-valued outcomes. Numerical experiments demonstrate that the proposed method yields valid bounded estimates for both compositional and distributional data, confirming its feasibility and practical utility in complex data environments.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

In applied research, Lee (2009) bounds are widely applied to bound the average treatment effect in the presence of selection bias. This paper extends the methodology of Lee bounds to accommodate outcomes in a general metric space, such as compositional and distributional data. By exploiting a representation of the Fr\'echet mean of the potential outcome via embedding in an Euclidean or Hilbert space, we present a feasible characterization of the identified set of the causal effect of interest, and then propose its analog estimator and bootstrap confidence region. The proposed method is illustrated by numerical examples on compositional and distributional data.

Problem

• Lee bounds
• average treatment effect
• selection bias
• metric space
• Fréchet mean

Innovation

• Lee bounds
• Fréchet mean
• metric space
• compositional data
• distributional data