Median Radial Function: A Robust, Covariance-Free Framework and Applications

TL;DR

This paper introduces a median-radius framework for robust, covariance-free assessment of centrality and dispersion in multivariate data, effective in high-dimensional and complex distributions.

Contribution

It proposes a novel, scale-invariant radial dispersion measure and depth function that are robust to outliers and adaptable to skewness, multimodality, and heavy tails.

Findings

The depth function is robust and independent of covariance structure.

Subgradients reveal data imbalance and asymmetry.

Empirical results match classical methods under symmetry, outperforming in complex data.

Abstract

A median-radius framework for assessing centrality in multivariate data using median distances is proposed. Based on the proposed framework, a scale invariant measure of radial dispersion is defined and used to establish a depth function that is robust to outliers and independent of covariance structure. The depth function does not depend on moment assumptions and naturally adapts to skewness, multimodality, and heavy-tailed distributions, which make it effective for high-dimensional data structures. We demonstrate fundamental characteristics of the underlying functionals such as subgradient and convexity. The subgradients provide additional insight and encode the imbalance in directional contributions of the data. This suggests a new approach to detect skewness and structural asymmetry through a purely radial construction. Empirical studies demonstrate that the method agrees with classical approaches under symmetry while providing a more flexible and informative characterization in complex settings.