Abstract
This is a paper about the foundation of robust inference. As a specific example, we consider semiparametric location models that involve a shape parameter. We argue that robust methods result via the selection of a representative shape from a set of allowable shapes. To perform this selection, we need a measure of disparity between the true shape and the shape to be used in the inference. Given such a disparity, we propose to solve a certain minimax problem. The paper discusses in detail the use of the Kullback-Leibler divergence for the selection of shapes. The resulting estimators are shown to have redescending influence functions when the set of allowable shapes contains heavy-tailed members. The paper closes with a brief discussion of the next logical step, namely the representation of a set of shapes by a pair of selected shapes.
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Morgenthaler, S., Hurvich, C. An information-theoretic framework for robustness. Ann Inst Stat Math 43, 131–146 (1991). https://doi.org/10.1007/BF00116473
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DOI: https://doi.org/10.1007/BF00116473