{"paper":{"title":"Median Radial Function: A Robust, Covariance-Free Framework and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A median radial depth function measures centrality in multivariate data without covariance or moment assumptions.","cross_cats":[],"primary_cat":"stat.ME","authors_text":"Elsayed Elamir","submitted_at":"2026-05-13T12:36:00Z","abstract_excerpt":"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 a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The depth function is robust to outliers and independent of covariance structure; it does not depend on moment assumptions and naturally adapts to skewness, multimodality, and heavy-tailed distributions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That median distances from an appropriately chosen center can serve as the basis for a statistically valid and useful depth function without additional moment or distributional assumptions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A new median radial function defines a covariance-free, outlier-robust depth measure for multivariate data that adapts to skewness and multimodality.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A median radial depth function measures centrality in multivariate data without covariance or moment assumptions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"26df95e4595b9472ff234db6af024df900afc007370bf1a2d40a56020b5c5693"},"source":{"id":"2605.13439","kind":"arxiv","version":1},"verdict":{"id":"48b0dc3f-c282-43d2-b0f8-7955d7426d3a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:11:28.192095Z","strongest_claim":"The depth function is robust to outliers and independent of covariance structure; it does not depend on moment assumptions and naturally adapts to skewness, multimodality, and heavy-tailed distributions.","one_line_summary":"A new median radial function defines a covariance-free, outlier-robust depth measure for multivariate data that adapts to skewness and multimodality.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That median distances from an appropriately chosen center can serve as the basis for a statistically valid and useful depth function without additional moment or distributional assumptions.","pith_extraction_headline":"A median radial depth function measures centrality in multivariate data without covariance or moment assumptions."},"references":{"count":2,"sample":[{"doi":"10.1007/s40300-","year":2021,"title":"Boente, G., & Salibián-Barrera, M. (2021). Robust functional principal components for sparse longitudinal data. METRON, 79(2), 159–188. https://doi.org/10.1007/s40300- 020-00193-3 Boyd, S., & Vandenbe","work_id":"18b6df31-7a6e-4f3a-9917-71cf7110e9e7","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s42519-021-00236-6","year":2019,"title":"On the Generalised Distance in Statistics","work_id":"ff7b2508-d3e8-42d2-b18a-7d56f5606fdd","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":2,"snapshot_sha256":"e6b607da50e84ba1f2b36d0b58fc192f37819f8fb7f4e8574364ec8b5462a769","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}