Optimal in uence curves for general loss functions

Generalizing MSE-optimality on 1/ √ n -shrinking neighborhoods of contamination type, we determine the robust in uence curve that minimizes maximum asymptotic risk, where risk may be any convex and isotone function G of L2 and L∞ -norms. The solutions necessarily minimize the trace of the covariance subject to an upper bound on the sup-norm, and also include an implicit equation for the optimal bound. For parameter dimension p = 1 , also the asymptotic minimax problem for neighborhoods of total variational type is solved. In technical respects, general risk may be reduced to MSE by weighting bias suitably. In case p = 1 , the result covers Lq -risks, q ∈ [1,∞) , con dence intervals of minimal length, and over-/undershooting probabilities. In case p > 1 , in addition to the L∞ norm, a solution for coordinatewise norms is given (relevant for total variation, p > 1 ). Passing to the least favorable contamination radius as in [RKR01], we obtain that for a large class of risks, the radius-minimax procedure does not depend on the function G .