AISM 53, 244-261
© 2001 ISM
(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.
Source ( TeX , DVI )