High Breakdown-Point and High Efficiency Robust Estimates for Regression

A class of robust estimates for the linear model is introduced. These estimates, called MM-estimates, have simultaneously the following properties: (i) they are highly efficient when the errors have a normal distribution and (ii) their breakdown-point is 0.5. The MM-estimates are defined by a three-stage procedure. In the first stage an initial regression estimate is computed which is consistent robust and with high breakdown-point but not necessarily efficient. In the second stage an M-estimate of the errors scale is computed using residuals based on the initial estimate. Finally, in the third stage an M-estimate of the regression parameters based on a proper redescending psi-function is computed. Consistency and asymptotical normality of the MM-estimates assuming random carriers are proved. A convergent iterative numerical algorithm is given. Finally, the asymptotic biases under contamination of optimal bounded influence estimates and MM-estimates are compared.