Two Moments for Risk-Monotone Additive Statistics

2026-08-11

AI Summary

This study characterizes the structure of statistical functionals that satisfy additivity and mean-preserving extended monotonicity under independent risks. By integrating tools from probability theory, convex analysis, and functional equations, the authors prove that within the space of distributions possessing finite p-th moments, such functionals are determined solely by the first moment when \( p < 2 \), and by non-negative linear combinations of the first and second moments when \( p \geq 2 \). This result not only establishes the necessity of variance in the case \( p \geq 2 \) but also unifies and extends classical risk measure theory, providing a complete axiomatic characterization of admissible statistical functionals under the stated conditions.

Abstract

Every statistic on laws with finite pth moment that is additive across independent risks and monotone in mean-preserving spreads depends only on an additive function of the mean when p is strictly less than 2, and only on such a function and a nonnegative multiple of the variance when p is no less than 2.

Problem

• additive statistics
• risk monotonicity
• mean-preserving spreads
• moments
• variance

Innovation

• additive statistics
• risk monotonicity
• mean-preserving spreads
• variance dependence
• moment conditions