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AN INFORMATION-THEORETIC FRAMEWORK FOR ROBUSTNESS

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STEPHAN MORGENTHALER^{1} AND CLIFFORD HURVICH^{2}

^{1} *Swiss Federal Institute of Technology, EPFL-DMA, 1015 Lausanne, Switzerland*

^{2} *New York University, 735 Tisch Hall, Washington Sq., New York, NY 10003, U.S.A.*
(Received December 27, 1988; revised July 13, 1989)

**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.

*Key words and phrases*:
Robustness, distributional shapes,
Kullback-Leibler divergence.

**Source**
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