## 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
Full Text: