Tail Structure and the Ordering of the Standard Deviation and Gini Mean Difference

Paper: arXiv 2601.12414

Authors: Nawaf Mohammed

Abstract

We investigate the ordering between two fundamental measures of dispersion for real-valued risks: the standard deviation (SD) and the Gini mean difference (GMD). Our analysis is driven by a single structural object, namely the mean excess function of the pairwise difference $|X - X’|$. We show that its monotonicity is determined by the tail behavior of the underlying distribution, giving rise to two distinct dispersion regimes. In a heavy-tailed regime, characterized by decreasing hazard rates or increasing reverse hazard rates, the SD dominates the GMD. Conversely, when both tails of the distribution are light, the GMD dominates the SD. These dominance regimes are shown to be stable under truncation, convolution, and mixtures. Discrete analogues of the main results are also developed. Overall, the results provide an intuitive interpretation of the dispersion ordering phenomena that goes beyond the existing general comparisons, with direct relevance for risk modeling and actuarial applications.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical exploration of the ordering between Standard Deviation and Gini Mean Difference using mean excess functions and hazard rates. While mathematically solid and clearly written, it is purely theoretical and entirely lacks empirical evaluation or backtesting on real-world financial data.

Research Flowchart

  flowchart TD
    A["Research Question: How do distribution tails affect SD vs. GMD ordering?"] --> B["Core Object: Analyze Mean Excess Function of pairwise difference |X - X'|"]
    
    B --> C["Heavy-Tailed Regime: Decreasing hazard rates / Increasing reverse hazard rates"]
    B --> D["Light-Tailed Regime: Light behavior in both tails"]
    
    C --> E["Finding 1: Standard Deviation (SD) dominates GMD"]
    D --> F["Finding 2: Gini Mean Difference (GMD) dominates SD"]
    
    E & F --> G["Properties & Applications: Orderings stable under truncation, convolution, mixtures; applied to risk/actuarial modeling"]