The cost of not knowing the radius

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 [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 sca...

• Type: Journal
• Year: 2008
• Where: SMA
• Authors: Helmut Rieder, Matthias Kohl, Peter Ruckdeschel
• Keywords: Ideal Models, Relative Risk, Robust Estimator, SMA 2008, Solid Modeling