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Optimal influence curves for general loss functions

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  • Ruckdeschel Peter
  • Rieder Helmut

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.

Suggested Citation

  • Ruckdeschel Peter & Rieder Helmut, 2004. "Optimal influence curves for general loss functions," Statistics & Risk Modeling, De Gruyter, vol. 22(3/2004), pages 201-223, March.
    • Handle: RePEc:bpj:strimo:v:22:y:2004:i:3/2004:p:201-223:n:3
    DOI: 10.1524/stnd.22.3.201.57067
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    References listed on IDEAS

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    1. Rieder, Helmut & Kohl, Matthias & Ruckdeschel, Peter, 2001. "The costs of not knowing the radius," SFB 373 Discussion Papers 2001,81, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
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    Cited by:

    1. Matthias Kohl & Peter Ruckdeschel & Helmut Rieder, 2010. "Infinitesimally Robust estimation in general smoothly parametrized models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 19(3), pages 333-354, August.
    2. Helmut Rieder & Matthias Kohl & Peter Ruckdeschel, 2008. "The cost of not knowing the radius," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 17(1), pages 13-40, February.

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