TAIL BEHAVIOR AND BREAKDOWN PROPERTIES OF EQUIVARIANT ESTIMATORS OF LOCATION

JOZEF KUŠNIER 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 breakdownrelated 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.