🤖 AI Summary
This study addresses the challenge of simultaneously achieving robustness and geometric interpretability in multivariate data analysis by proposing the Moving Median Absolute Deviation (MMAD) depth function, which introduces the median absolute deviation into a statistical depth framework for the first time. Built upon a median absolute distance functional, MMAD characterizes the arrangement of observations within the central 50% region through directional derivatives, gradient representations, and spherical boundary distributions, enabling efficient computation without complex optimization or projections. The method not only aligns with classical depth approaches in identifying central observations but also uncovers directional geometric features of the data. Furthermore, it establishes a theoretical connection to robust measures of skewness, offering both practical utility and enhanced interpretability in robust multivariate analysis.
📝 Abstract
We propose and analyze the moving median absolute deviation (MMAD) as a robust depth construction based on the median absolute distance functional with particular emphasis on its local geometry and probabilistic structure. In the univariate setting, we derive the derivative of the MMAD scale and interpret it through boundary mass imbalance, thereby establishing a direct connection to a robust skewness measure. This idea extends naturally to a multivariate setting that describes how observations are arranged along the 50% central region using a directional derivative, a gradient representation, and a spherical boundary distribution. From a computational perspective, MMAD can be estimated efficiently using distance calculations without needing complex optimization or projection schemes. Multivariate applications based on depth correlations, contour visualizations, and central region overlap demonstrate that MMAD identifies essentially the same central observations as classical depth notions while delivering additional information and geometric insight about directional structure. These features make MMAD a practical and informative approach for robust multivariate data analysis.