The least trimmed squares. I: Consistency. (English) Zbl 1248.62033

Summary: The consistency of the least trimmed squares estimator [see P.J. Rousseeuw, J. Am. Stat. Assoc. 79, 871–880 (1984; Zbl 0547.62046); or F.R. Hampel et al., Robust statistics. The approach based on influence functions. NY etc.: Wiley and Sons (1986; Zbl 0593.62027)] is proved under general conditions. The assumptions employed in this paper are discussed in detail to clarify the consequences for applications.


62F12 Asymptotic properties of parametric estimators
62J05 Linear regression; mixed models
62F35 Robustness and adaptive procedures (parametric inference)
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[1] Andrews D. W. K.: Consistency in nonlinear econometric models: A generic uniform law of large numbers. Econometrica 55 (1987), 1465-1471 · Zbl 0646.62101 · doi:10.2307/1913568
[2] Bickel P. J.: One-step Huber estimates in the linear model. J. Amer. Statist. Assoc. 70 (1975), 428-433 · Zbl 0322.62038 · doi:10.2307/2285834
[3] Breiman L.: Probability. Addison-Wesley, London 1968 · Zbl 0753.60001
[4] Chatterjee S., Hadi A. S.: Sensitivity Analysis in Linear Regression. Wiley, New York 1988 · Zbl 0648.62066
[5] Csőrgő M., Révész P.: Strong Approximation in Probability and Statistics. Akademia Kiadó, Budapest 1981 · Zbl 0539.60029
[6] Dhrymes P. J.: Introductory Econometrics. Springer-Verlag, New York 1978 · Zbl 0388.62096
[7] Jurečková J., Sen P. K.: Regression rank scores scale statistics and studentization in linear models. Proc. Fifth Prague Symposium on Asymptotic Statistics, Physica Verlag, Heidelberg 1993, pp. 111-121
[8] Hampel F. R., Ronchetti E. M., Rousseeuw P. J., Stahel W. A.: Robust Statistics - The Approach Based on Influence Functions. Wiley, New York 1986 · Zbl 0733.62038
[9] Hettmansperger T. P., Sheather S. J.: A cautionary note on the method of least median squares. Amer. Statist. 46 (1992), 79-83
[10] Liese F., Vajda I.: Consistency of \(M\)-estimators in general models. J. Multivar. Anal. 50 (1994), 93-114 · Zbl 0872.62071 · doi:10.1006/jmva.1994.1036
[11] Portnoy S.: Tightness of the sequence of empiric c. d.f. processes defined from regression fractiles. Robust and Nonlinear Time-Series Analysis (J. Franke, W. Härdle, and D. Martin, Springer-Verlag, New York 1983, pp. 231-246
[12] Prigogine I., Stengers I.: La Nouvelle Alliance. SCIENTIA 1977, Issues 5-12
[13] Prigogine I., Stengers I.: Out of Chaos. William Heinemann Ltd, London 1984
[14] Rousseeuw P. J.: Least median of square regression. J. Amer. Statist. Assoc. 79 (1984), 871-880 · Zbl 0547.62046 · doi:10.2307/2288718
[15] Rousseeuw P. J., Leroy A. M.: Robust Regression and Outlier Detection. Wiley, New York 1987 · Zbl 0711.62030
[16] Rubio A. M., Víšek J. Á.: A note on asymptotic linearity of \(M\)-statistics in nonlinear models. Kybernetika 32 (1996), 353-374 · Zbl 0882.62053
[17] Rubio A. M., Víšek J. Á.: Estimating the contamination level of data in the framework of linear regression analysis. Qüestiió 21 (1997), 9-36 · Zbl 1167.62388
[18] Štěpán J.: Teorie pravděpodobnosti (Probability Theory). Academia, Prague 1987
[19] Huffel S. Van: Total least squares and error-in-variables modelling: Bridging the gap between statistics, computational mathematics and enginnering. Proc. Computational Statistics, COMPSTAT 2004 (J. Antoch, Physica-Verlag, Springer 2004, pp. 539-555 · Zbl 1153.62335
[20] Víšek J. Á.: On high breakdown point estimation. Comput. Statistics 11 (1996), 137-146 · Zbl 0933.62015
[21] Víšek J. Á.: Sensitivity analysis \(M\)-estimates. Ann. Inst. Statist. Math. 48 (1996), 469-495 · Zbl 0925.62131 · doi:10.1007/BF00050849
[22] Víšek J. Á.: Ekonometrie I (Econometrics I). Carolinum, Publishing House of Charles University, Prague 1997
[23] Víšek J. Á.: Robust specification test. Proc. Prague Stochastics’98 (M. Hušková, P. Lachout, and J. Á. Víšek, Union of Czechoslovak Mathematicians and Physicists, Prague 1998, pp. 581-586
[24] Víšek J. Á.: Robust instruments. Robust’98 (J. Antoch and G. Dohnal, Union of Czechoslovak Mathematicians and Physicists, Prague 1998, pp. 195-224
[25] Víšek J. Á.: Robust estimation of regression model. Bull. Czech Econometric Society 9 (1999), 57-79
[26] Víšek J. Á.: The least trimmed squares - random carriers. Bull. Czech Econometric Society 10 (1999), 1-30
[27] Víšek J. Á.: The robust regression and the experiences from its application on estimation of parameters in a dual economy. Proc. Macromodels’99, Rydzyna 1999, pp. 424-445
[28] Víšek J. Á.: On the diversity of estimates. Comput. Statist. Data Anal. 34 (2000) 67-89 · Zbl 1052.62509 · doi:10.1016/S0167-9473(99)00068-7
[29] Víšek J. Á.: Regression with high breakdown point. Robust 2000 (J. Antoch and G. Dohnal, Union of Czechoslovak Mathematicians and Physicists, Prague 2001, pp. 324-356
[30] Víšek J. Á.: A new paradigm of point estimation. Proc. Data Analysis 2000/II, Modern Statistical Methods - Modelling, Regression, Classification and Data Mining (K. Kupka, TRYLOBITE, Pardubice 2000, 195-230
[31] Víšek J. Á.: Sensitivity analysis of \(M\)-estimates of nonlinear regression model: Influence of data subsets. Ann. Inst. Statist. Math. 54 (2002), 2, 261-290 · Zbl 1013.62072 · doi:10.1023/A:1022465701229
[32] Víšek J. Á.: \(\sqrt{n}\)-consistency of empirical distribution function of residuals in linear regression model. Probab. Lett., submitted
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