ROBUSTNESS OF THE HALF-SPACE MEDIAN

In this note, we extend Donoho and Gasko's (Ann. Statist. 20 (1992) 1803-1827) results on finite sample breakdown point for the half-space median to any given data set and, as a consequence, we obtain bounds for the limiting breakdown point for general (in particular, nonsymmetric) distributions. Lower and upper bounds for the breakdown point of the half-space median with respect to the half-space metric are also established; in two-dimensional space, they yield the exact breakdown point. The exact 'gross error neighborhood' breakdown point for symmetric distributions is also given.