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

Nawaf Mohammed

Papers from arXiv.org

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.

Date: 2026-01, Revised 2026-01
References: Add references at CitEc
Citations:

Downloads: (external link)
http://arxiv.org/pdf/2601.12414 Latest version (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:2601.12414

Access Statistics for this paper

More papers in Papers from arXiv.org
Bibliographic data for series maintained by arXiv administrators ().