The cost of not knowing the radius

Robust Statistics considers the quality of statistical decisions in the presence of deviations from the ideal model. Usually these deviations are neighborhoods of a certain size about the ideal model. We introduce a new concept of optimality if this size or radius is not (precisely) known: We determine the increase of the maximum risk over the minimax risk in the case that the optimally robust estimator for the false neighborhood radius is used. The maximum increase of the relative risk is minimized in the case that the radius is known only to belong to some interval [rl, ru] . We pursue this minmax approach for a number of ideal models and a variety of neighborhoods. Also, the effect of increasing parameter dimension is studied for these models. The minimax increase of relative risk in case the radius is completely unknown, 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 ρ ≤ 1 3 , in order to keep the increase of relative risk below 12.5% , provided that 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 .