A Robust Moment System Based on Absolute Deviations and Quantile Slicing

TL;DR

This paper introduces robust quantile-sliced moment systems based on absolute deviations and medians, providing reliable tools for distributional analysis especially in heavy-tailed or contaminated data scenarios.

Contribution

It develops MAD and MedAD moments that are robust, exist under broad conditions, and unify distributional inference methods for heavy-tailed and contaminated data.

Findings

MAD moments are efficient for light to moderate tails.

MedAD moments are stable even when higher moments do not exist.

MedAD estimators offer robust alternatives to likelihood-based methods.

Abstract

This study develops two robust, quantile-sliced moment systems, mean and median absolute deviation (MAD and MedAD moments), to serve as foundational tools in parametric modeling, statistical inference, and describing distributional location, scale, skewness, and tail behavior in settings where classical moments and L-moments fail. MAD moments use block-wise absolute deviations around the median and exist whenever the mean is finite, while MedAD moments replace expectations with medians, ensuring existence for all distributions, including heavy-tailed cases with undefined mean or variance. The systems exhibit strong consistency, slice-based robustness, and bounded influence. The results indicate that MAD and L moment ratios are efficient for light to moderate tails, whereas MedAD ratios remain uniquely stable when higher moments do not exist. Applications to Cauchy parameter estimation highlight the practical value of MedAD estimators as simple, fully robust alternatives to likelihood-based approaches. Together, these systems offer a unified, median-anchored framework for reliable distributional inference under heavy tails and contamination.