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

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.