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Abstract

The Gross-Error Sensitivity (GES) and the Breakdown Point (BP) are two measures of quantitative robustness which have played a key role in the development of the theory of robust-ness. Both can be derived from the maximum bias function B(€) and constitute a two-number summary of this function. The GES is the derivative of B(€) at the origin whereas the BP determines the asymptote of the curve (c, B(€)). Since GES€ ≈ B(€) for € near zero, the GES summarizes the behavior of B(€) near the origin. On the other hand, the BP does not provide an approximation for B(€) for c large and, consequently, estimates with strikingly different bias performance when c is large may have the same BP. A new robustness quantifier, the breakdown rate (BR), that summarizes the behavior of B(€)for € near BP will be introduced. The BR for several families of robust estimates of regression will be presented and the increased usefulness of the three-number summary (GES,BP,BR) for comparing robust estimates will be illustrated by several examples.