AISM 53, 244-261
© 2001 ISM

Tail behavior and breakdown properties of equivariant estimators of location

Jozef Kusnier and Ivan Mizera

Department of Statistics, University of Illinois at Urbana-Champaign, 101 Illini Hall, 725 South Wright Street, Champaign, IL 61820, U.S.A.

(Received May 13, 1998; revised August 23, 1999)

Abstract.    For translation and scale equivariant estimators of location, inequalities connecting tail behavior and the finite-sample breakdown point are proved, analogous to those established by He et al. (1990, Econometrika, 58, 1195-1214) for monotone and translation equivariant estimators. Some other inequalities are given as well, enabling to establish refined bounds and in some cases exact values for the tail behavior under heavy- and light-tailed distributions. The inequalities cover translation and scale equivariant estimators in great generality, and they involve new breakdown-related quantities, whose relations to the breakdown point are discussed. The worth of tail-behavior considerations in robustness theory is demonstrated on examples, showing the impact of the basic two techniques in robust estimation: trimming and averaging. The mathematical language employs notions from regular variation theory.

Key words and phrases:    Robustness, breakdown, tail behavior, equivariance, location estimator, regular variation.

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