🤖 AI Summary
This study addresses the limitations of classical and L-moments in characterizing distributional features under heavy-tailed or contaminated data, where they often fail to provide robust inference. The authors propose a novel robust moment system—MAD moments and MedAD moments—anchored at the median, which integrates absolute deviations, quantile slicing, and median-based expectations to establish a unified framework for robust statistical inference. These moments are well-defined for any distribution, including heavy-tailed cases lacking finite means, and possess bounded influence functions along with slice-wise robustness. Theoretical analysis and empirical experiments demonstrate that MAD moments achieve high efficiency under light-to-moderate tails, while MedAD moments remain stable even when higher-order moments do not exist. Notably, in parameter estimation for the Cauchy distribution, the proposed methods significantly outperform conventional likelihood-based approaches, offering both robustness and practical utility.
📝 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.