Statistical Methods and Applications

, Volume 17, Issue 1, pp 13–40 | Cite as

The cost of not knowing the radius

Original Article


Robust Statistics considers the quality of statistical decisions in the presence of deviations from the ideal model, where deviations are modelled by neighborhoods of a certain size about the ideal model. We introduce a new concept of optimality (radius-minimaxity) if this size or radius is not precisely known: for this notion, 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 [r l ,r u ]. 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% versus 57.1% and 50.5% versus 172.1% for one-dimensional location and scale, respectively, and less than 1/3 in other typical contamination models. In most models considered so far, the radius needs to be specified only up to a factor \(\rho\le \frac{1}{3}\), 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.


Symmetric location and contamination Infinitesimal asymmetric neighborhoods Total variation, contamination Asymptotically linear estimators Influence curves Maximum asymptotic variance and mean square error Relative risk Inefficiency Least favorable radius Radius–minimax robust estimator Location, scale, regression models 

Mathematics Subject Classification (2000)

62F35 62G35 62G05 62J05 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Helmut Rieder
    • 1
  • Matthias Kohl
    • 1
  • Peter Ruckdeschel
    • 1
  1. 1.Department of MathematicsUniversity of BayreuthBayreuthGermany

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