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

Battle Count: 55

The paper is primarily relevant to risk modeling and actuarial applications rather than direct trading strategy development. However, the SD-GMD ordering has implications for: (1) selection of appropriate dispersion measures in portfolio risk assessment; (2) understanding when quadratic risk measures (variance-based) overstate or understate variability relative to linear measures; (3) tail risk characterization relevant to heavy-tailed asset returns; (4) the Gini-based risk framework referenced in Furman et al. (2017) and Chen et al. (2025) for portfolio selection. The results are more foundational/theoretical than directly implementable in trading algorithms, but inform the choice of risk metrics in quantitative finance.

As a purely theoretical paper, there is no algorithm or model to implement. The practical application involves: (1) computing hazard and reverse hazard rates for a given distribution to check monotonicity conditions; (2) verifying log-concavity of the density via second derivative of log-density; (3) computing SD and GMD numerically for specific distributions. These are straightforward numerical tasks. The theoretical proofs themselves require advanced knowledge of inequalities (Chebyshev, Prékopa-Leindler, Hölder) and stochastic ordering theory.

The paper is entirely theoretical with complete proofs provided in Appendix A (12 sub-proofs). All numerical examples include explicit formulas for SD and GMD. The analytical framework is self-contained with classical inequalities (Chebyshev, Prékopa-Leindler, Hölder) referenced to standard texts. No empirical data or code is required for verification. All distributional examples use well-known parametric families with closed-form or numerically tractable expressions.