Optimal influence curves for general loss functions

Ruckdeschel Peter and Rieder Helmut

Statistics & Risk Modeling, 2004, vol. 22, issue 3, 201-223

Abstract: 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 Lq-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.

Date: 2004
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DOI: 10.1524/stnd.22.3.201.57067

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