Breakdown Points of Affine Equivariant Estimators of Multivariate Location and Covariance Matrices
Open accessAbstract
Finite-sample replacement breakdown points are derived for different types of estimators of multivariate location and covariance matrices. The role of various equivariance properties is illustrated. The breakdown point is related to a measure of performance based on large deviations probabilities. Finally, we show that one-step reweighting preserves the breakdown point.
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If at least (1-α)k points lie within distance r of z, then any geometric median x* satisfies ||x*-z||_2 ≤ C_α r, where C_α = (1-α)/sqrt(1-2α).
“If |{j : ∥x j -z∥ 2 ≤ r}| ≥ (1 -α)k, then x * -z 2 ≤ C α r, C α ≜ 1 -α √ 1 -2α . (16”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Lopuhaä and Rousseeuw (1991)
Theorem 2.2 and its proof (p. 5) establish that the L1 estimator, which minimizes the sum of distances, remains bounded when fewer than half the points are replaced. The proof gives a different bound and does not state the claimed radius bound or the constant Cα = (1−α)/√(1−2α). (LLM, conf=0.82)
Experimental — accuracy not yet measuredThe 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 Lopuhaä and Rousseeuw, 1991
The finite-sample breakdown point of the geometric median is ε* = ⌈N/2⌉/N, which tends to 1/2 as N→∞ and is optimal among translation-equivariant estimators.
“Breakdown point. The finite-sample breakdown point is ϵ ⋆ = ⌈N/2⌉/N → 1/2 as N → ∞, which is optimal among translation-equivariant estimators (Lopuhaä and Rousseeuw, 1991).”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Lopuhaä and Rousseeuw, 1991
The geometric median has breakdown point exactly ⌈N/2⌉/N, which tends to 1/2 as N→∞.
“Hence the breakdown point is exactly ϵ ⋆ = ⌈N/2⌉/N , which tends to 1/2 as N → ∞.”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Lopuhaä and Rousseeuw, 1991
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Research works printed but not linked: 8
- DoNOHo, D. L. (1982). Breakdown properties of multivariate location estimators. Ph.D. dissertation, Harvard Univ.
- DoNOHO, D. L. and HuBER, P. J. (1983). The notion of breakdown point. In A Festschrift for Erich L. Lehmann (P. J. Bickel, K. A. Doksum and J. L. Hodges, Jr., eds.) 157–184. Wadsworth, Belmont, Calif.
- HAmPEL, F. R. (1968). Contributions to the theory of robust estimation. Ph.D. dissertation, Univ. California, Berkeley.
- HoDGEs, J. L., JR. (1967). Efficiency in normal samples and tolerance of extreme values for some estimates of location. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 163-186. Univ. California Press, Berkeley.
- RoussEEUw, P. J. (1985). Multivariate estimation with high breakdown point. In Mathematical Statistics and Applications (W. Grossmann, G. Pflug, I. Vincze and W. Wertz, eds.) 8 283-297. Reidel, Dordrecht.
- RoussEEUw, P. J. and LEROy, A. M. (1987). Robust Regression and Outlier Detection. Wiley, New York.
- RoussÉEUw, P. J. and YoHAI, V. (1984). Robust regression by means of S estimators. Robust and Nonlinear Time Series Analysis. Lecture Notes in Statist. 26 256-272. Springer, New York.
- STAHEL, W. A. (1981). Robust estimation: Infinitesimal optimality and covariance matrix estimators. Ph.D. dissertation, ETH, Zurich. (In German.)
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Shi, W., Ling, Q., Wu, G., & Yin, W. (2015) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2015.2461520 · Cited by 307 · 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
- Lopuhaa, H. P., & Rousseeuw, P. J.
- ORCID
- Rousseeuw, Peter J.
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 1991-03-01
- Volume
- 19
- Issue
- 1
- Pages
- 229-248
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- OpenAlex
- W2136179595
- Cited by
- 446
- References
- 0
- Citations by year
- 2026: 9 · 2025: 15 · 2024: 15 · 2023: 20 · 2022: 14 · 2021: 13 · 2020: 12 · 2019: 14 · 2018: 30 · 2017: 23 · 2016: 19 · 2015: 17 · 2014: 19 · 2013: 15 · 2012: 13
- FWCI
- 8.76
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Research Repository (Delft University of Technology) (submittedVersion)
- Data Archiving and Networked Services (DANS) (submittedVersion, other-oa)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Keywords
- breakdown point, affine equivariance, multivariate location, covariance matrix estimation, large deviations, robust estimation
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