PreprintCC BY 4.0arXiv
One-Cut Risk Profiles under Quadratic Loss: Discrete Convexity, Continuous Limits, and Higher Dimensions
Abstract
In this note we study a two-regime representation of a loss random variable under quadratic error. For a finite law we compute exactly the change of the optimal risk when one atom crosses the cut. This turns the problem into a convexity question in cumulative-mass coordinates. On an equally spaced support, log-concavity gives this convexity, while weak symmetry locates the optimal cut, with an additional correction when the mean lies between two atoms. We also discuss the continuous analogue and extend the main identities to finitely supported random vectors, where a global optimal partition may be chosen as a halfspace.
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BibTeX
@misc{nistor2026onecut,
title = {{One-Cut Risk Profiles under Quadratic Loss: Discrete Convexity, Continuous Limits, and Higher Dimensions}},
author = {Mihaela-Adriana Nistor and Ionel Popescu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05357},
eprint = {2609.05357},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05357},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - One-Cut Risk Profiles under Quadratic Loss: Discrete Convexity, Continuous Limits, and Higher Dimensions
AU - Mihaela-Adriana Nistor
AU - Ionel Popescu
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05357
UR - https://arxiv.org/abs/2609.05357
AB - In this note we study a two-regime representation of a loss random variable under quadratic error. For a finite law we compute exactly the change of the optimal risk when one atom crosses the cut. This turns the problem into a convexity question in cumulative-mass coordinates. On an equally spaced support, log-concavity gives this convexity, while weak symmetry locates the optimal cut, with an additional correction when the mean lies between two atoms. We also discuss the continuous analogue and extend the main identities to finitely supported random vectors, where a global optimal partition may be chosen as a halfspace.
ER -