Skewness and Kurtosis for Heavy-Tailed Distributions: A Comparison of Mean Absolute Deviation and L-Moment Approaches for Pareto and Extreme-Value Families

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.