Abstract
For the linear regression model we define SM-estimators with the following properties:
With high probability the estimator is identical with the least squares estimator, if the data correspond to the assumption of normality. Especially, the estimation procedure has 95% asymptotic efficiency at the normal model.
The procedure will identify groups of data which do not follow the model, as long as more than 50% of the data are in correspondence with the ideal model. Especially, the SM-estimator has breakdown point 1/2.
We evaluate bounds on the supremum asymptotic bias (SAB) for rather big balls around the ideal model, which will prove uniform consistency, too.
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References
Behnen, K. (1991). Robuste Statistik: Multiple Lineare Regression. Lecture Notes: Mathem. Stochastik, Univ. Hamburg 1991
Dudley, R.M. (1979). Balls in ℝk do not cut all subsets of k + 2 points. Adv. Math. 31 (1979), pp. 306–308.
Hennig, Chr. (1993). Efficient high-breakdown-estimators in robust regression. Preprint: Mathem. Stochastik, Univ. Hamburg 1993
Huber, P.J. (1981). Robust Statistics. Wiley: New York 1981.
Maronna, R.A. & Yohai, V.J. (1981). Asymptotic behavior of general M-estimates for regression and scale with random carriers. Z. Wahrsch. verw. Geb. 58 (1981), pp. 7–20.
Martin, R.D., Yohai, V.J., Zamar, R.H. (1989). Min-Max bias robust regression. Ann. Statist. 17 (1989), pp. 1608–1630.
Rousseeuw, P.J. & Leroy, A.M. (1987). Robust Regression and Outlier Detection. Wiley: New York 1987.
Rousseeuw, P.J. & Yohai, V.J. (1984). Robust Regression by Means of S-estimators. In J. Franke, W. Härdle & D. Martin (Eds.), Robust and NonlinearTime Series Analysis, pp. 256–272, Springer: Berlin 1984.
Shorack, G.R. & Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley: New York 1986.
Yohai, V.J. (1987). High breakdown-point and high efficiency robust estimates for regression. Ann. Statist. 15 (1987), pp. 642–656.
Yohai, V.J. & Zamar, R. (1988). High breakdown point estimates of regression. J. Amer. Statist. Assoc. 83 (1988), pp. 406–413.
Yohai, V.J., Stahel, W.A., Zamar, R.H. (1991). A procedure for robust estimation and inference in linear regression. In W.Stahel & S.Weisberg (Eds.), Directions in Robust Statistics and Diagnostics, Part 2, pp. 365–374, Springer: New York 1991.
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© 1994 Springer-Verlag Berlin Heidelberg
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Behnen, K. (1994). A Modification of Least Squares with High Efficiency and High Breakdown Point in Linear Regression. In: Mandl, P., Hušková, M. (eds) Asymptotic Statistics. Contributions to Statistics. Physica, Heidelberg. https://doi.org/10.1007/978-3-642-57984-4_14
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DOI: https://doi.org/10.1007/978-3-642-57984-4_14
Publisher Name: Physica, Heidelberg
Print ISBN: 978-3-7908-0770-7
Online ISBN: 978-3-642-57984-4
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