Breakdown points of t-type regression estimators

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 1/2 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.