发表机构
Faculty of Mathematics and Computer Science, University of Bucharest; Institute of Mathematics “Simion Stoilow” of the Romanian Academy（布加勒斯特大学数学与计算机科学学院; 罗马尼亚科学院西米恩·斯托伊洛瓦数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对二次损失下的单割风险轮廓问题，将其转化为累积质量坐标下的凸性问题，讨论了连续类比并扩展至高维随机向量的最优划分，为相关风险分析提供了理论支撑。
AI 中文摘要
本文研究二次误差下损失随机变量的两区域表示。对于有限律，我们精确计算了当一个原子越过割点时最优风险的变化，这将问题转化为累积质量坐标下的凸性问题。在等距支撑上，对数凹性给出该凸性，而弱对称性确定最优割点，当均值位于两个原子之间时存在额外修正。我们还讨论了连续类比，并将主要恒等式扩展到有限支撑的随机向量，其中全局最优划分可选取为半空间。
英文摘要
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.
