The Annals of Statistics

The "Automatic" Robustness of Minimum Distance Functionals

David L. Donoho and Richard C. Liu

Full-text: Open access

Abstract

The minimum distance (MD) functional defined by a distance $\mu$ is automatically robust over contamination neighborhoods defined by $\mu$. In fact, when compared to other Fisher-consistent functionals, the MD functional was no worse than twice the minimum sensitivity to $\mu$-contamination, and at least half the best possible breakdown point. In invariant settings, the MD functional has the best attainable breakdown point against $\mu$-contamination among equivariant functionals. If $\mu$ is Hilbertian (e.g., the Hellinger distance), the MD functional has the smallest sensitivity to $\mu$-contamination among Fisher-consistent functionals. The robustness of the MD functional is inherited by MD estimates, both estimates based on "weak" distances and estimates based on "strong" distances, when the empirical distribution is appropriately smoothed. These facts are general and apply not just in simple location models, but also in multivariate location-scatter and in semiparametric settings. Of course, this robustness is formal because $\mu$-contamination neighborhoods may not be large enough to contain realistic departures from the model. For the metrics we are interested in, robustness against $\mu$-contamination is stronger than robustness against gross errors contamination; and for "weak" metrics (e.g., $\mu = \text{Cramer-von Mises, Kolmogorov})$, robustness over $\mu$-neighborhoods implies robustness over Prohorov neighborhoods.

Article information

Source
Ann. Statist. Volume 16, Number 2 (1988), 552-586.

Dates
First available: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176350820

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aos/1176350820

Mathematical Reviews number (MathSciNet)
MR947562

Zentralblatt MATH identifier
0684.62030

Subjects
Primary: 62F35: Robustness and adaptive procedures
Secondary: 62F12: Asymptotic properties of estimators

Keywords
Quantitative robustness gross-error sensitivity breakdown point Kolmogorov Levy Prohorov variation Hellinger distances Cramer-von Mises discrepancy

Citation

Donoho, David L.; Liu, Richard C. The "Automatic" Robustness of Minimum Distance Functionals. The Annals of Statistics 16 (1988), no. 2, 552--586. doi:10.1214/aos/1176350820. http://projecteuclid.org/euclid.aos/1176350820.


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