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 MAD-integral shape metrics built from the quantile function, paired with simple quadrature rules, provide interpretable, robust distributional shape analysis and estimation across heavy-tailed and non-Gaussian settings?

Methodology

Derive MAD-based shape functionals as quantile integrals; analyze midpoint quadrature recovery of classical quantile metrics; design and evaluate C-Trapezoid quadrature; compare approximation error across standard families; derive closed-form parameter estimates; analyze breakdown properties vs MLE; present two end-to-end case studies.

Key findings

• (Under Review)

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