Proofs that the Gerber Statistic is Positive Semidefinite

ArXiv ID: 2305.05663 “View on arXiv”

Authors: Unknown

Abstract

In this brief note, we prove that both forms of the Gerber statistic introduced in Gerber et al. (2022) are positive semi-definite.

Keywords: Gerber Statistic, Positive Semi-Definite, Risk Management, Dependence Modeling, General (Risk Measurement)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is dense with advanced linear algebra proofs, demonstrating matrix transformations and series expansions to establish positive semidefiniteness, which is a purely theoretical property with no practical implementation details provided. It contains no backtesting, datasets, or statistical metrics, focusing solely on the mathematical validity of the Gerber statistic.

  flowchart TD
    A["Research Goal<br/>Prove Gerber Statistic is PSD"] --> B["Analyze Structure<br/>1-form and 2-form"]
    B --> C["Mathematical Derivation<br/>Matrix Factorization & Boundaries"]
    C --> D["Computational Verification<br/>Symbolic/Numerical Analysis"]
    D --> E["Key Findings<br/>Both forms are Positive Semi-Definite"]
    E --> F["Outcomes<br/>Validated for Risk Management & Dependence Modeling"]