High breakdown-point and high efficiency robust estimates for regression. (English) Zbl 0624.62037

Yohai, Victor J.

Ann. Stat. 15, 642-656 (1987).

Author’s summary: 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.

Reviewer: J.Antoch

MSC:

62F35	Robustness and adaptive procedures (parametric inference)
62J05	Linear regression; mixed models

Keywords:

high efficiency; robust estimates; linear model; MM-estimates; three- stage procedure; initial regression estimate; high breakdown-point; M- estimate of the errors scale; residuals; M-estimate of the regression parameters; redescending psi-function; Consistency; asymptotical normality; iterative numerical algorithm; asymptotic biases; contamination; optimal bounded influence estimates