Optimal Allocation and Volume under Surface
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.
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BibTeX
@misc{feng2026optimal,
title = {{Optimal Allocation and Volume under Surface}},
author = {Kai Feng and Han Hong and Jessie Li and Wenshi Wei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38875},
eprint = {2609.38875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38875},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal Allocation and Volume under Surface
AU - Kai Feng
AU - Han Hong
AU - Jessie Li
AU - Wenshi Wei
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38875
UR - https://arxiv.org/abs/2609.38875
AB - 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.
ER -