Open Access
March, 1992 Robust Direction Estimation
Xuming He, Douglas G. Simpson
Ann. Statist. 20(1): 351-369 (March, 1992). DOI: 10.1214/aos/1176348526


We relate various measures of the stability of estimates in general parametric families and consider their application to direction estimates on spheres. We show that constructions such as the SB-robustness of Ko and Guttorp and the information-standardized gross-error sensitivity of Hampel, Ronchetti, Rousseeuw and Stahel fit into a general framework in which one measures the effect of model contamination by the Kullback-Leibler discrepancy. We also define a breakdown point appropriate for a compact parameter space. Specific results concerning direction estimation include the optimal robustness of the circular median, the optimal breakdown point of the least median of squares on the sphere, the SB-robustness of certain scale-adjusted $M$-estimators and the SB-robustness in arbitrary dimensions of a class of estimators including the $L_1$-estimator and the hyperspherical median. The latter estimators avoid the need for simultaneous scale estimates, and they have breakdown points approaching 1/2 as the model becomes concentrated. A slight modification in their definition yields the same theoretical breakdown point as the least median of squares.


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Xuming He. Douglas G. Simpson. "Robust Direction Estimation." Ann. Statist. 20 (1) 351 - 369, March, 1992.


Published: March, 1992
First available in Project Euclid: 12 April 2007

zbMATH: 0761.62035
MathSciNet: MR1150348
Digital Object Identifier: 10.1214/aos/1176348526

Primary: 62F35
Secondary: 62F12

Keywords: Breakdown function , Breakdown point , directional mean , least median of squares , SB-robustness , spherical median , von Mises distribution

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 1 • March, 1992
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