Berrendero, José R.; Mendes, Beatriz V. M.; Tyler, David E. On the maximum bias functions of \(MM\)-estimates and constrained \(M\)-estimates of regression. (English) Zbl 1114.62030 Ann. Stat. 35, No. 1, 13-40 (2007). Summary: We derive the maximum bias functions of the \(MM\)-estimates and the constrained \(M\)-estimates or \(CM\)-estimates of regression and compare them to the maximum bias functions of the \(S\)-estimates and the \(\tau\)-estimates of regression. In these comparisons, the \(CM\)-estimates tend to exhibit the most favorable bias-robustness properties. Also, under the Gaussian model, it is shown how one can construct a \(CM\)-estimate which has a smaller maximum bias function than a given \(S\)-estimate, that is, the resulting \(CM\)-estimate dominates the \(S\)-estimate in terms of maxbias and, at the same time, is considerably more efficient. Cited in 9 Documents MSC: 62F35 Robustness and adaptive procedures (parametric inference) 62J05 Linear regression; mixed models 62F10 Point estimation Keywords:robust regression; M-estimates; S-estimates; constrained M-estimates; maximum bias curves; breakdown point; gross error sensitivity; method of moments estimates × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Berrendero, J. R. and Zamar, R. (2001). Maximum bias curves for robust regression with non-elliptical regressors. Ann. Statist. 29 224–251. · Zbl 1029.62028 · doi:10.1214/aos/996986507 [2] 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 [3] He, X. (1991). A local breakdown property of robust tests in linear regression. J. Multivariate Anal. 38 294–305. · Zbl 0736.62028 · doi:10.1016/0047-259X(91)90047-6 [4] He, X. and Simpson, D. G. (1993). Lower bounds for contamination bias: Globally minimax versus locally linear estimation. Ann. Statist. 21 314–337. · Zbl 0770.62023 · doi:10.1214/aos/1176349028 [5] Hennig, C. (1995). Efficient high-breakdown-point estimators in robust regression: Which function to choose? Statist. Decisions 13 221–241. · Zbl 0844.62026 [6] Maronna, R., Bustos, O. and Yohai, V. J. (1979). Bias- and efficiency-robustness of general \(M\)-estimators for regression with random carriers. In Smoothing Techniques for Curve Estimation . Lecture Notes in Math. 757 91–116. Springer, Berlin. · Zbl 0416.62050 · doi:10.1007/BFb0098492 [7] Martin, R. D., Yohai, V. J. and Zamar, R. H. (1989). Min-max bias robust regression. Ann. Statist. 17 1608–1630. · Zbl 0713.62068 · doi:10.1214/aos/1176347384 [8] Mendes, B. V. M. and Tyler, D. E. (1996). Constrained \(M\)-estimation for regression. In Robust Statistics, Data Analysis and Computer Intensive Methods . Lecture Notes in Statist. 109 299–320. Springer, New York. · Zbl 0839.62031 [9] Rousseeuw, P. J. (1984). Least median of squares regression. J. Amer. Statist. Assoc. 79 871–880. JSTOR: · Zbl 0547.62046 · doi:10.2307/2288718 [10] Rousseeuw, P. J. and Yohai, V. J. (1984). Robust regression by means of S-estimators. In Robust and Nonlinear Time Series Analysis . Lecture Notes in Statist. 26 256–272. Springer, New York. · Zbl 0567.62027 [11] Yohai, V. J. (1987). High breakdown-point and high efficiency robust estimates for regression. Ann. Statist. 15 642–656. · Zbl 0624.62037 · doi:10.1214/aos/1176350366 [12] 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 · doi:10.2307/2288856 [13] Yohai, V. J. and Zamar, R. H. (1993). A minimax-bias property of the least \(\alpha\)-quantile estimates. Ann. Statist. 21 1824–1842. · Zbl 0797.62027 · doi:10.1214/aos/1176349400 [14] Yohai, V. J. and Zamar, R. H. (1997). Optimal locally robust \(M\)-estimates of regression. J. Statist. Plann. Inference 64 309–323. · Zbl 0914.62024 · doi:10.1016/S0378-3758(97)00040-2 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.