Analyzing distortion riskmetrics and weighted entropy for unimodal and symmetric distributions under partial information constraints

Paper: arXiv 2504.19725

Authors: Baishuai Zuo, Chuancun Yin

Abstract

In this paper, we develop the lower and upper bounds of worst-case distortion riskmetrics and weighted entropy for unimodal, and symmetric unimodal distributions when mean and variance information are available. We also consider the sharp upper bounds of distortion riskmetrics and weighted entropy for symmetric distribution under known mean and variance. These results are applied to (weighted) entropies, shortfalls and other risk measures. Specifically, entropies include cumulative Tsallis past entropy, cumulative residual Tsallis entropy of order α, extended Gini coefficient, fractional generalized cumulative residual entropy, and fractional generalized cumulative entropy. Shortfalls include extended Gini shortfall, Gini shortfall, shortfall of cumulative residual entropy, and shortfall of cumulative residual Tsallis entropy. Other risk measures include nth-order expected shortfall, dual power principle and proportional hazard principle.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous, advanced mathematical proofs deriving worst-case bounds for distortion riskmetrics under partial information and shape constraints. However, it is purely theoretical with minimal empirical validation or real-data backtesting provided in the excerpt.

Research Flowchart

  flowchart TD
    A[Research Goal: Bound distortion riskmetrics & weighted entropy under partial information] --> B[Inputs: Mean, Variance & Structural Constraints]
    B --> C[Constraints: Unimodal, Symmetric Unimodal, & Symmetric Distributions]
    C --> D[Methodology: Derive Sharp Upper & Lower Bounds via Optimization]
    D --> E[Applications: Calculate Risk Measures & Entropies]
    E --> F[Findings: Bounds for Tsallis/Gini Entropies, Extended Shortfalls & ES]