Yohai, Victor J.; Zamar, Ruben H. High breakdown-point estimates of regression by means of the minimization of an efficient scale. (English) Zbl 0648.62036 J. Am. Stat. Assoc. 83, No. 402, 406-413 (1988). 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. Cited in 4 ReviewsCited in 100 Documents MSC: 62F35 Robustness and adaptive procedures (parametric inference) 62F10 Point estimation 62F12 Asymptotic properties of parametric estimators 62E20 Asymptotic distribution theory in statistics Keywords:bias robustness; high efficiency; tau estimators; adaptive weights; new class of robust estimates; breakdown point; normal errors; M estimate; consistency; asymptotic normality; convergent iterative computing algorithm; biases; gross error contamination; optimal bounded-influence estimates PDFBibTeX XMLCite \textit{V. J. Yohai} and \textit{R. H. Zamar}, J. Am. Stat. Assoc. 83, No. 402, 406--413 (1988; Zbl 0648.62036) Full Text: DOI