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

TL;DR

This paper explores how the tail behavior of distributions determines whether the standard deviation or Gini mean difference is larger, revealing two distinct regimes with implications for risk modeling.

Contribution

It introduces a novel tail-based framework to compare dispersion measures, establishing stability of these regimes under various distributional operations.

Findings

Heavy tails lead to SD dominance over GMD.

Light tails result in GMD dominance over SD.

Regimes are stable under truncation, convolution, and mixtures.

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.