A class of robust and fully efficient regression estimators. (English) Zbl 1012.62073

Summary: This paper introduces a new class of robust estimators for the linear regression model. They are weighted least squares estimators, with weights adaptively computed using the empirical distribution of the residuals of an initial robust estimator. It is shown that under certain general conditions the asymptotic breakdown points of the proposed estimators are not less than that of the initial estimator, and the finite sample breakdown point can be at most \(1/n\) less. For the special case of the least median of squares as initial estimator, hard rejection weights and normal errors and carriers, the maximum bias function of the proposed estimators for point-mass contaminations is numerically computed, with the result that there is almost no worsening of bias.
Moreover – and this is the original contribution of this paper – if the errors are normally distributed and under fairly general conditions on the design the proposed estimators have full asymptotic efficiency. A Monte Carlo study shows that they have better behavior than the initial estimators for finite sample sizes.


62J05 Linear regression; mixed models
62F35 Robustness and adaptive procedures (parametric inference)
62H12 Estimation in multivariate analysis
Full Text: DOI


[1] ADROVER, J., BIANCO, A. and YOHAI, V. J. (2001). Approximate -estimates for linear regression based on subsampling of elemental sets. In Statistics in Genetics and in the Environmental Sciences (L. T. Fernholz, S. Morgenthaler and W. Stahel, eds.) 141-151. Birkäuser, Basel.
[2] AGOSTINELLI, C. and MARKATOU, M. (1998). A one-step robust estimator for regression based on the weighted likelihood reweighting scheme. Statist. Probab. Lett. 37 341-350. · Zbl 0902.62077
[3] DAVIES, L. (1990). The asymptotics of S-estimators in the linear regression model. Ann. Statist. 18 1651-1675. · Zbl 0719.62042
[4] DONOHO, D. L. and HUBER, P. J. (1983). The notion of breakdown point. In A Festschrift for Erich L. Lehmann (P. J. Bickel, K. A. Doksum and J. L. Hodges, Jr., eds.) 157-184. Wadsworth, Belmont, CA. · Zbl 0523.62032
[5] GERVINI, D. and YOHAI, V. J. (2000). A class of robust and fully efficient regression estimators. Unpublished manuscript. · Zbl 1012.62073
[6] HAMPEL, F. R. (1975). Beyond location parameters: robust concepts and methods (with discussion). Bull. ISI 46 375-391. · Zbl 0349.62029
[7] HAMPEL, F. R., RONCHETTI, E. M., ROUSSEEUW, P. J. and STAHEL, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley, New York. · Zbl 0593.62027
[8] HE, X. and PORTNOY, S. (1992). Reweighted LS estimators converge at the same rate as the initial estimator. Ann. Statist. 20 2161-2167. · Zbl 0764.62043
[9] HÖSSJER, O. (1992). On the optimality of S-estimators. Statist. Probab. Lett. 14 413-419. · Zbl 0761.62036
[10] MARTIN, R. D., YOHAI, V. J. and ZAMAR, R. H. (1989). Min-max bias robust regression. Ann. Statist. 17 1608-1630. · Zbl 0713.62068
[11] POLLARD, D. (1990). Empirical Processes: Theory and Applications. IMS, Hayward, CA. · Zbl 0741.60001
[12] ROUSSEEUW, P. J. (1984). Least median of squares regression. J. Amer. Statist. Assoc. 79 871-880. JSTOR: · Zbl 0551.62049
[13] ROUSSEEUW, P. J. and LEROY, A. (1987). Robust Regression and Outlier Detection. Wiley, New York. · Zbl 0711.62030
[14] ROUSSEEUW, P. J. and YOHAI, V. (1984). Robust regression by means of S-estimators. Robust and Nonlinear Time Series Analysis. Lecture Notes in Statist. 26 256-272. Springer, New York. · Zbl 0567.62027
[15] SIMPSON, D. G. and YOHAI, V. J. (1998). Functional stability of one-step GM-estimators in approximately linear regression. Ann. Statist. 26 1147-1169. · Zbl 0930.62030
[16] YOHAI, V. J. (1987). High breakdown-point and high efficiency robust estimates for regression. Ann. Statist. 15 642-656. · Zbl 0624.62037
[17] YOHAI, V. J. and ZAMAR, R. H. (1988). High breakdown-point estimates of regression by means of the minimization of an efficient scale. J. Amer. Statist. Assoc. 83 406-413. JSTOR: · Zbl 0648.62036
[18] YOHAI, V. J. and ZAMAR, R. H. (1993). A minimax-bias property of the least -quantile estimates. Ann. Statist. 21 1824-1842. · Zbl 0797.62027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.