Abstract
In this paper we show that approximate τ-estimates for the linear model, computed by the algorithm based on subsampling of elemental subsets, are consistent and with high probability have the same breakdown point that the exactτ-estimate. Then, if these estimates are used as initial values, the reweighted least squares algorithm yields a local minimum of the τ-scale having the same asymptotic distribution and, with high probability, the same breakdown point that the global minimum.
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References
Agulló, J. (1996).Computation of Estimates with High Breakdown Point.PhDthesis, University of Alicante, Faculty of Economics. (In Spanish).
Donoho, D. L. and Huber, P. J. (1983). The notion of breakdown point. InP. J. Bickel, K. L. D. and Jr., J. L. H., editorsA Festschrift for Erich L.Lehmann, pages 157–184. Wadsworth, Belmont, California.
Hampel, F. R. (1971). A general qualitative definition of robustness. Ann.Math. Statist.42:1887–1896.
Hawkins, D. M. (1993a). The accuracy of elemental set approximations for regression.J. Amer. Statist. Assoc.88:580–589.
Hawkins, D. M. (1993b). The feasible set algorithm for least median of squares regression.Computational Statistics and Data Analysis16:81–101.
Hössjer, O. (1992). On the optimality of s—estimators.Statistics and Probability Letters14:413–419.
Huber, P. J. (1981).Robust Statistics.John Wiley, New York.
Maronna, R. and Yohai, V. J. (1993). Bias—robust estimates of regression based on projections. Ann.Statist.21:965–990.
Rousseeuw, P. (1984). Least median of squares regression.J. Amer. Statist. Assoc.79:871–880.
Rousseeuw, P. and Basset, G. (1991). Robustness of the p—subset algorithm for regression with high breakdown point. In Stahel, W. and Weisberg, S., editorsDirections in Robust Statistics and Diagnostics Part II. IMA vols. in Math. and its Appl.volume 34, pages 185–194. Springer, New York.
Rousseeuw, P. J. and Yohai, V. J. (1984). Robust regression by means of s—estimators. In J. Franke, W. H. and Martin, R. D., editorsRobust and Nonlinear Time Series Analysisvolume 26 of Lecture Notes in Statisticpages 256–272. Springer—Verlag, New York.
Stromberg, A. J. (1993). Computing the exact least median of squares estimate and stability diagnostics in multiple linear regression.SIAM J. Sci. Comput.14:1289–1299.
Yohai, V. J. and Zamar, R. (1988). High breakdown point estimates of regression by means of the minimization of an efficient scale.J. Amer. Statist. Assoc.83:406–413.
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Adrover, J., Bianco, A., Yohai, V. (2001). Approximate τ—Estimates for Linear Regression Based on Subsampling of Elemental Sets. In: Fernholz, L.T., Morgenthaler, S., Stahel, W. (eds) Statistics in Genetics and in the Environmental Sciences. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8326-9_12
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DOI: https://doi.org/10.1007/978-3-0348-8326-9_12
Publisher Name: Birkhäuser, Basel
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Online ISBN: 978-3-0348-8326-9
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