Optimal Allocation and Volume under Surface

2026-09-29

AI Summary

This study addresses the challenges of estimating and conducting inference for the volume under the ROC surface (VUS) by constructing a theoretical framework based on Aumann expectation and Minkowski mixed volumes, and proposes a doubly debiased machine learning estimator. The core innovation lies in establishing, for the first time, a connection between VUS and symmetric U-statistics, thereby enabling unbiased inference in high-dimensional settings. This work rigorously derives the asymptotic properties of the proposed estimator and develops valid inference procedures. Furthermore, it resolves fundamental issues related to VUS computation and inequality measurement, effectively extending the applicability boundaries of the Gini coefficient.

Abstract

This paper develops a framework for estimation and inference on the volumes of sets that are projections of critical function sets, focusing particularly on the convex body beneath the optimal receiver operating characteristic (ROC) surface. Specifically, we propose a volume calculation method that first uses an Aumann expectation representation and then applies Minkowski mixed volumes. Using this framework, we show that the population volume under the ROC surface (VUS) is proportional to the expectation of a symmetric U-statistic kernel. We then propose a double/debiased machine learning estimator of the VUS, derive its asymptotic properties, and develop an inference procedure. Further applications of this framework include an analysis of the feasible error set across pre-defined groups and a natural generalization of the Gini coefficient for measuring inequality.

Problem

• Volume under Surface
• ROC surface
• Optimal allocation
• Estimation and inference
• Convex body

Innovation

• Volume under ROC surface
• Minkowski mixed volumes
• Double/debiased machine learning
• U-statistic
• Aumann expectation