Skewness and Kurtosis for Heavy-Tailed Distributions: A Comparison of Mean Absolute Deviation and L-Moment Approaches for Pareto and Extreme-Value Families
- 1Department of Computer Science, Metropolitan College, Boston University, 1010 Commonwealth Avenue, Boston, MA 02215, USA
Abstract
Classical skewness and kurtosis require finite third and fourth moments and are therefore undefined for Pareto-type and extreme-value models with small tail index. We study shape measures based on the mean absolute deviation (MAD), which require only a finite mean, and ask how they compare with L-moment ratios in interpretation, computation, and parameter estimation. For the Pareto I–IV, Fréchet, Gumbel, and Weibull families, we derive closed-form expressions, verified by numerical quadrature, for the MAD about any quantile, for MAD skewness and kurtosis, and for the corresponding L-moment ratios, using incomplete beta and gamma functions. The MAD measures are bounded and have a simple interpretation in terms of sub-means. We then propose MAD-Q, which matches theoretical and empirical MAD at the quartiles, gives its Jacobian in closed form, and characterizes identifiability: with known scale, the estimating equations have a unique solution, whereas with unknown scale, the MAD ratio is not monotone and a quartile-consistency step is required. In Monte Carlo experiments with known scale, MAD-Q is the most accurate non-likelihood method for most families and remains usable under lower-tail contamination that breaks the Pareto I likelihood; with unknown scale, its advantage is confined to infinite-variance tails. Financial loss and wind-speed applications illustrate both the strengths and limitations of the approach.
Related
- Comparison of two approaches to weighted topological entropy
- Zero Flux: Flow-Based Comparison of High-Dimensional Discrete Distributions
- Efficient Closure Learning in Dynamical Systems: A Comparison of Data-Assimilated and Embedded Approaches
- Estimating heterogeneous treatment effects from randomised trials: a comparison of the risk modelling and treatment effect modelling approaches
- Unsupervised Careless Responding Detection: A Comprehensive Comparison between Model-based and Model-Agnostic Approaches
BibTeX
@article{pinsky2026skewness,
title = {{Skewness and Kurtosis for Heavy-Tailed Distributions: A Comparison of Mean Absolute Deviation and L-Moment Approaches for Pareto and Extreme-Value Families}},
author = {Eugene Pinsky and Triparna Kundu},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203656},
url = {https://www.mdpi.com/2227-7390/14/20/3656},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Skewness and Kurtosis for Heavy-Tailed Distributions: A Comparison of Mean Absolute Deviation and L-Moment Approaches for Pareto and Extreme-Value Families
AU - Eugene Pinsky
AU - Triparna Kundu
PY - 2026
DA - 2026/10/09
JO - Mathematics
DO - 10.3390/math14203656
UR - https://www.mdpi.com/2227-7390/14/20/3656
AB - Classical skewness and kurtosis require finite third and fourth moments and are therefore undefined for Pareto-type and extreme-value models with small tail index. We study shape measures based on the mean absolute deviation (MAD), which require only a finite mean, and ask how they compare with L-moment ratios in interpretation, computation, and parameter estimation. For the Pareto I–IV, Fréchet, Gumbel, and Weibull families, we derive closed-form expressions, verified by numerical quadrature, for the MAD about any quantile, for MAD skewness and kurtosis, and for the corresponding L-moment ratios, using incomplete beta and gamma functions. The MAD measures are bounded and have a simple interpretation in terms of sub-means. We then propose MAD-Q, which matches theoretical and empirical MAD at the quartiles, gives its Jacobian in closed form, and characterizes identifiability: with known scale, the estimating equations have a unique solution, whereas with unknown scale, the MAD ratio is not monotone and a quartile-consistency step is required. In Monte Carlo experiments with known scale, MAD-Q is the most accurate non-likelihood method for most families and remains usable under lower-tail contamination that breaks the Pareto I likelihood; with unknown scale, its advantage is confined to infinite-variance tails. Financial loss and wind-speed applications illustrate both the strengths and limitations of the approach.
SN - 2227-7390
ER -