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

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.