Robust Direction Estimation

Abstract

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.