First, we show that many robust estimates of regression which depend only on the regression residuals (including M-, S-, Tau-, least median of squares-, least trimmed of squares-and some R-estimates) have infinite gross-error-sensitivity. More precisely, we show that the maximumbias function of a large class of estimates, called residual admissible in Yohai and Zamar (Ann. Statist. :21, 1993, 1824-1842), is of order v/~ near zero. Based on this finding we define a new robustness measure for estimates with Br(~)= o(~/~), the contamination sensitivity of order /3, which extends Hampel's gross error sensitivity for estimates with unbounded influence. We compute this measure for regression M-estimates with a general scale and show that ]~ = 0.5 in this case. Then we solve a Hampel-like optimality problem, namely, one of minimizing the asymptotic variance subject to a bound on the contamination sensitivity of order fl = 0.5, for estimates in this class. Finally, we show that a certain least ~-quantile estimate has the smallest contamination sensitivity of order 0.5 among all residual admissible estimates. In the Gaussian case c~ =0.683. (~) 1997 Elsevier Science B.V.