A Survey of Multidimensional Medians
Abstract
In this paper we survey a sequence of papers whose primary aim is the generalization of the concept of the median into higher dimensional settings. While a variety of distinct definitions of the median of a multivariate data set are possible these definitions have the common property of producing the usual definition when applied to univariate data or a univariate distribution. Some common ideas of equivariance, symmetry and breakdown are discussed as well as computational convenience for each definition. The extension of these ideas to directional statistics is also discussed.
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The half-space (Tukey) median attains the optimal breakdown 1/2 in any dimension but is NP-hard to compute and prohibitive at d ≥ 5.
“the half-space (Tukey) median attains the optimal breakdown 1/2 in any dimension but is NP-hard to compute and prohibitive at d ≥ 5 (Small, 1990)”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Small, 1990
The geometric median attains the optimal 1/2 breakdown for any translation-equivariant estimator.
“The geometric median attains the optimal 1/2 breakdown for any translation-equivariant estimator (Lopuhaä and Rousseeuw, 1991;Small, 1990;Vardi and Zhang, 2000)”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Small, 1990
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Robust statistics in data analysis — A review
Daszykowski, M., Kaczmarek, K., Vander Heyden, Y., & Walczak, B. (2007) · Chemometrics and Intelligent Laboratory Systems · doi:10.1016/j.chemolab.2006.06.016 · Cited by 331 · see citing works
RoPoLL: Robust Panel of LLM Judges
Acharya, A., Pan, K. W., & Verkhovsky, B. (2026) · arXiv · doi:10.48550/arxiv.2606.30931
- Authors
- Small, C. G.
- Venue
- International Statistical Review / Revue Internationale de Statistique
- Publisher
- JSTOR
- Published
- 1990-12
- Volume
- 58
- Issue
- 3
- Pages
- 263
- ISSN
- 0306-7734
- Cited by
- 278
- References
- 0
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