Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness
Open accessAbstract
We describe multivariate generalizations of the median, trimmed mean and $W$ estimates. The estimates are based on a geometric construction related to "projection pursuit." They are both affine equivariant (coordinate-free) and have high breakdown point. The generalization of the median has a breakdown point of at least $1/(d + 1)$ in dimension $d$ and the breakdown point can be as high as $1/3$ under symmetry. In contrast, various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than $1/(d + 1)$ in dimension $d$.
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The breakdown floor is matched up to a dimensional constant only by the (intractable) Tukey halfspace median.
“the breakdown floor, in turn, is matched up to a dimensional constant only by the (intractable) Tukey halfspace median.”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Donoho and Gasko, 1992
The opening summary (p. 1) gives the halfspace-depth median a breakdown point of at least 1/(d + 1), and §5.3 (p. 13) says the paper’s estimators are computationally impractical in higher dimensions. But “only” is not supported: Proposition 3.5 (p. 9) establishes high breakdown for an outlyingness-weighted mean, and §5.2 (p. 12) describes another method with breakdown nearly 1/2. (LLM, conf=0.87)
Experimental — accuracy not yet measuredThe minimax-optimal estimator on the breakdown floor is the Tukey halfspace median.
“The minimax-optimal estimator on the breakdown floor is the Tukey halfspace median (Tukey, 1975;Donoho and Gasko, 1992)”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Donoho and Gasko, 1992
The introduction (p. 1) identifies the halfspace-depth generalization of the median and gives its breakdown-point lower bound of 1/(d + 1). It does not establish that this estimator is minimax-optimal; the discussion of breakdown results (p. 9) instead says the outlyingness-weighted mean has better breakdown properties and is nearly best possible. (LLM, conf=0.84)
Experimental — accuracy not yet measuredThe minimax-optimal estimator on the breakdown floor is the Tukey halfspace median, whose exact computation is NP-hard for d ≥ 3; the smoothed-depth estimator of Chen et al. (2018) matches the σα floor in sub-exponential time.
“The minimax-optimal estimator on the breakdown floor is the Tukey halfspace median (Tukey, 1975;Donoho and Gasko, 1992), whose exact computation is NP-hard for d ≥ 3 (Johnson and Preparata, 1978;Aloupis, 2006); the smoothed-depth estimator of Chen et al. (2018) matches the σα floor in sub-exponential time.”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Donoho and Gasko, 1992
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Research works printed but not linked: 10
- DoNoHo, D. L. and HuBER, P. J. (1982). The notion of breakdown point. In A Festschrift for Erich L. Lehmann in Honor of His Sixty-fifth Birthday (P. J. Bickel, K. A. Doksum and J. L. Hodges, Jr., eds.) 157–184. Wadsworth, Belmont, Calif.
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- RoussEEUw, P. (1985). Multivariate estimation with high breakdown point. In Mathematical Statistics and Applications, Vol. B (W. Grossman, G. Ch. Pflug, I. Vincze and W. Wertz, eds.) 283–297. Reidel, Dordrecht.
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RoPoLL: Robust Panel of LLM Judges
Acharya, A., Pan, K. W., & Verkhovsky, B. (2026) · arXiv · doi:10.48550/arxiv.2606.30931
- Authors
- Donoho, D. L., & Gasko, M.
- ORCID
- Donoho, David L.
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 1992-12-01
- Volume
- 20
- Issue
- 4
- Pages
- 1803-1827
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- OpenAlex
- W2039892753
- Cited by
- 583
- References
- 0
- Citations by year
- 2026: 6 · 2025: 23 · 2024: 16 · 2023: 30 · 2022: 28 · 2021: 35 · 2020: 31 · 2019: 20 · 2018: 24 · 2017: 36 · 2016: 24 · 2015: 21 · 2014: 19 · 2013: 21 · 2012: 27
- FWCI
- 3.39
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- 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 › Computer Science › Artificial Intelligence › Anomaly Detection Techniques and Applications
- Keywords
- halfspace depth, breakdown point, multivariate location estimation, projection pursuit, affine equivariance, trimmed mean, robust statistics, outliers
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