Depth-Based Vector Median Absolute Deviation Moments for Robust Multivariate Shape Analysis

TL;DR

This paper introduces a robust, median-based approach for multivariate shape analysis using vector median absolute deviation moments, improving robustness and interpretability over classical methods.

Contribution

It proposes VMedAD moments that replace traditional covariance-based moments with median-based contrasts derived from data depth, offering affine invariance and robustness.

Findings

VMedAD moments are affine equivariant and moment-free.

They effectively separate central structure from tail-driven behavior.

Simulation and real data show improved robustness and interpretability.

Abstract

Classical multivariate shape analysis relies on covariance-standardized moments, such as Mardia skewness and kurtosis, which are sensitive to outliers and require finite moments. This paper introduces vector median absolute deviation (VMedAD) moments for robust multivariate shape analysis. The proposed framework replaces moment aggregation and covariance standardization with median-based center-outward contrasts defined through data depth, yielding affine equivariance and moment-free vector moments. VMedAD moments provide direction-preserving measures of multivariate skewness and directional peripheral dominance, separating central structure from tail-driven behavior. Consistency, breakdown properties, and affine equivariance are established, and simulation and real dataset examples demonstrate improved robustness and geometric interpretability over classical and projection-based methods.