© 2000 by Biometrika Trust
Breakdown points of t-type regression estimators
Department of Statistics, University of Illinois at Urbana-Champaign, 725 South Wright Street, Champaign, IL 61820, USA E-mail: x-he@uiuc.edu; dgs@uiuc.edu Z Capital One Financial Corporation, 11011 West Broad Street, Glen Allen, Virginia 23060, USA E-mail: guangyu.wang@capitalone.com
To bound the influence of a leverage point, generalised M-estimators have been suggested. However, the usual generalised M-estimator of regression has a breakdown point that is less than the inverse of its dimension. This paper shows that dimension-independent positive breakdown points can be attained by a class of well-defined generalised M-estimators with redescending scores. The solution can be determined through optimisation of t-type likelihood applied to properly weighted residuals. The highest breakdown point of 
is attained by Cauchy score. These bounded-influence and high-breakdown estimators can be viewed as a fully iterated version of the one-step generalised M-estimates of Simpson, Ruppert & Carroll (1992) with the two advantages of easier interpretability and avoidance of undesirable roots to estimating equations. Given the design-dependent weights, they can be computed via EM algorithms. Empirical investigations show that they are highly competitive with other robust estimators of regression.
Key Words: breakdown point; generalised M-estimator; linear regression; likelihood; robustness; t-distribution