The Annals of Statistics

High Breakdown-Point and High Efficiency Robust Estimates for Regression

Victor J. Yohai

Full-text: Open access


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.

Article information

Ann. Statist. Volume 15, Number 2 (1987), 642-656.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F35: Robustness and adaptive procedures
Secondary: 62J05: Linear regression

Linear model robust estimation high breakdown-point high efficiency


Yohai, Victor J. High Breakdown-Point and High Efficiency Robust Estimates for Regression. Ann. Statist. 15 (1987), no. 2, 642--656. doi:10.1214/aos/1176350366.

Export citation