Elementary and Robust Distribution Shape Analysis via Mean Absolute Deviations and Quantile-Based Quadrature Approximations

The crisis

• Heavy-tailed and variance-free distributions appear everywhere in risk, operations, and reliability — yet many estimators assume finite variance
• Shape summaries (skewness, kurtosis, tail weight) drive decisions; fragile estimators break under contamination
• Mean absolute deviation bridges quantiles and robustness: interpretable geometry without Gaussian assumptions
• Simple quadrature rules keep the methodology usable where analysts cannot run heavy iterative fitting

About this research

We connect quantile functions and mean absolute deviations by deriving MAD-based shape metrics expressed as integrals of the quantile function, with a direct geometric reading. The framework covers distributions with finite mean, including cases without finite variance (e.g. Pareto). With midpoint quadrature, the construction recovers standard quantile-shape summaries (interquartile range, Galton skewness, Moore octile kurtosis). We introduce a C-Trapezoid quadrature rule (cubic endpoint extrapolation plus trapezoidal integration) that sharply lowers approximation error versus the midpoint rule on common distributions, and yields closed-form, non-iterative estimation formulas where no closed-form CDF exists — with stronger outlier resilience than MLE (12.5% breakdown per tail). Two case studies illustrate quick distributional shape assessment without specialized tooling.

Research question

Can a single MAD-based family of distribution-shape metrics, expressed as integrals of the quantile function, recover the standard quantile-shape summaries as special cases — and can a simple cubic-endpoint quadrature rule approximate them accurately enough to yield closed-form, outlier-resilient parameter estimates for distributions that may lack a finite variance or an explicit CDF?

Methodology

MAD-based shape metrics are derived as integrals of the quantile function with a direct geometric (subarea) interpretation. Midpoint quadrature over octiles is shown to recover the interquartile range, Galton skewness, and Moore octile kurtosis as special cases. A C-Trapezoid rule (cubic-polynomial endpoint extrapolation combined with trapezoidal integration) is introduced and its approximation error compared against the midpoint and rectangle rules across common continuous distributions, including heavy-tailed cases without finite variance (e.g. Pareto). Closed-form, non-iterative octile-based estimators are derived and contrasted with maximum likelihood, quartile fitting, and L-moment estimation, and illustrated with two case studies.

Key findings

• MAD-based shape metrics expressed as integrals of the quantile function recover the interquartile range, Galton skewness, and Moore octile kurtosis as special cases under midpoint quadrature.
• The proposed C-Trapezoid rule (cubic endpoint extrapolation plus trapezoidal integration) achieves significantly lower approximation error than the midpoint rule across common distributions.
• It yields closed-form, non-iterative parameter-estimation formulas that apply even when no explicit CDF exists or the variance is infinite (e.g. Pareto).
• The estimators are more outlier-resilient than maximum likelihood (12.5% breakdown per tail) while retaining a simple geometric interpretation.
• Two case studies demonstrate quick distributional-shape assessment without specialized tooling: technology-sector ETF returns (XLK, 1999–2025) and weight distributions inside pretrained transformers.

References

• Galton (1882) · Inquiries into Human Faculty and Its Development
• Moore (1907) · The statistical complement of median, Journal of the American Statistical Association
• Huber & Ronchetti (2009) · Robust Statistics
• Code & artifacts · anacodicAI-labs/c-trapezoid-quantile-metrics