Extensions of the Mean Difference for the Lognormal Distribution

This paper extends the closed-form formula of Gini’s mean difference for the lognormal distribution, originally obtained by Girone and Manca (2016). The following are analyzed: 1) the asymptotic behavior of the scale parameter for γ → 0 (degeneration of the distribution) and for γ → +∞ (unbounded dispersion); 2) the generalized formula with complete location parameter μ and scale parameter σ; 3) the mean difference conditioned on an interval; 4) the mean difference for the truncated lognormal. Applications in the field of medical sciences are also discussed, where the lognormal distribution is ubiquitous in the modeling of biomarkers and pharmacological concentrations. The results show that the original formula Δ = 2eγ 2 2erf (γ 2) for the standardized case admits natural extensions that significantly broaden its field of application, while preserving the analytical elegance of the basic formulation.