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High breakdown-point estimates of regression by means of the minimization of an efficient scale. (English) Zbl 0648.62036

A new class of robust estimates, \(\tau\) estimates, is introduced. The estimates have simultaneously the following properties: (a) they are qualitatively robust, (b) their breakdown point is.5, and (c) they are highly efficient for regression models with normal errors. They are defined by minimizing a new scale estimate, \(\tau\), applied to the residuals. Asymptotically, a \(\tau\) estimate is equivalent to an M estimate with a \(\psi\) function given by a weighted average of two \(\psi\) functions, one corresponding to a very robust estimate and the other to a highly efficient estimate. The weights are adaptive and depend on the underlying error distribution. We prove consistency and asymptotic normality and give a convergent iterative computing algorithm. Finally, we compare the biases produced by gross error contamination in the \(\tau\) estimates and optimal bounded-influence estimates.

MSC:

62F35 Robustness and adaptive procedures (parametric inference)
62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
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