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Breakdown points of $t$-type regression estimators. (English) Zbl 1120.62320
Summary: 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 $\frac{1}{2}$; 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 and 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.
##### MSC:
 62J05 Linear regression 62F35 Robustness and adaptive procedures (parametric inference) 62F10 Point estimation