We determine the increase of the maximum risk over the minimax risk in the case that the optimally robust estimator for the false radius is used. This is done by numerical solution of the implicit equations which determine optimal robustness, for location, scale, and linear regression models, and by evaluation of maximum asymptotic variance and mean square error over fixed size symmetric contamination and infinitesimal asymmetric neighborhoods, respectively. The maximum increase of the relative risk is minimized in the case that the radius is known only to belong to some interval [pr, l'/p] The effect of increasing parameter dimension is studied for these models. The minimax increase of relative risk in ease p = 0, compared with that of the most robust procedure, is 18.1% vs. 57.1% and 50.5% vs. 172.1% for one-dimensional location and scale, respectively, and less than 1/3 in other typical contamination models. In most of our models, the radius needs to be specified only up to a factor p::; ~, in order to keep the increase of relative risk below 12.5%, provided and the radius minimax robust estimator is employed. The least favorable radii leading to the radius minimax estimators turn out small: 5% - 6% contamination, at sample size 100.
