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Peter Ruckdeschel, Helmut Rieder

Optimal influence curves for general loss functions

Generalizing MSE-optimality on 1/√n-shrinking neighborhoods of contamination type, we determine the robust influence 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 L_q-risks, q ∈ [1,∞), confidence 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.

Statistics & Decisions, Oldenbourg Wissenschaftsverlag

Print ISSN: 0721-2631
Volume: 22, 03/2004
Pages: 201 - 223

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