zbMATH — the first resource for mathematics

The least trimmed squares. III: Asymptotic normality. (English) Zbl 1248.62035
[For Part II see ibid., 181–202 (2006; Zbl 1248.62005).]
Summary: Asymptotic normality of the least trimmed squares estimator is proved under general conditions. At the end of paper a discussion of applicability of the estimator (including the discussion of algorithm for its evaluation) is offered.

62F12 Asymptotic properties of parametric estimators
62J05 Linear regression; mixed models
62F35 Robustness and adaptive procedures (parametric inference)
Full Text: Link
[1] Benáček V. J., Jarolím, M., Víšek J. Á.: Supply-side characteristics and the industrial structure of Czech foreign trade. Proc. Business and Economic Development in Central and Eastern Erupe: Implications for Economic Integration into wider Europe. Technical University in Brno and University of Wisconsin, Whitewater, and the Nottingham Trent University 1998, pp. 51-68
[2] Boček P., Lachout P.: Linear programming approach to LMS-estimation. Comput. Statist. Data Anal. (Memorial volume) 19(1995), 129-134 · Zbl 0875.62292 · doi:10.1016/0167-9473(93)E0051-5
[3] Breiman L.: Probability. Addison-Wesley Publishing Company, London 1968 · Zbl 0753.60001
[4] Chatterjee S., Hadi A. S.: Sensitivity Analysis in Linear Regression. Wiley, New York 1988 · Zbl 0648.62066
[5] Čížek P.: Robust estimation with discrete explanatory variables. COMPSTAT 2003, pp. 509-514
[6] Čížek P., Víšek J. Á.: Least trimmed squares. EXPLORE, Application Guide (W. Härdle, Z. Hlavka, and S. Klinke, Springer-Verlag, Berlin 2000, pp. 49-64
[7] Jurečková J., (1993) P. K. Sen: 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] Hawkins D. M., Olive D. J.: Improved feasible solution algorithms for breakdown estimation. Comput. Statist. Data Anal. 30 (1999), 1, 1-12 · Zbl 1042.62529 · doi:10.1016/S0167-9473(98)00082-6
[10] Hettmansperger T. P., Sheather S. J.: A cautionary note on the method of least median squares. Amer. Statist. 46 (1992), 79-83
[11] Huber P. J.: Robust Statistics. Wiley, New York 1981 · Zbl 0536.62025
[12] 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
[13] Maronna R. A., Yohai V. J.: Asymptotic behaviour of general \(M\)-estimates for regression and scale with random carriers. Z. Wahrscheinlichkeitstheorie verw. Gebiete 58 (1981), 7-20 · Zbl 0451.62031 · doi:10.1007/BF00536192
[14] Pollard D.: Asymptotics for least absolute deviation regression estimator. Econometric Theory 7 (1991), 186-199 · Zbl 04504753 · doi:10.1017/S0266466600004394
[15] 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
[16] Rousseeuw P. J., Leroy A. M.: Robust Regression and Outlier Detection. Wiley, New York 1987 · Zbl 0711.62030
[17] Rubio A. M., Víšek J. Á.: Estimating the contamination level of data in the framework of linear regression analysis. Qűestiió 21 (1997), 9-36 · Zbl 1167.62388 · www.idescat.cat · eudml:40196
[18] Štěpán J.: Teorie pravděpodobnosti (Probability Theory). Academia, Prague 1987
[19] Víšek J. Á.: A cautionary note on the method of Least Median of Squares reconsidered. Trans. Twelfth Prague Conference on Inform. Theory, Statist. Dec. Functions and Random Processes, Prague 1994, pp. 254-259
[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.: Diagnostics of regression subsample stability. Probab. Math. Statist. 17 (1997), 2, 231-257 · Zbl 0924.62072
[23] Víšek J. Á.: Robust estimation of regression model. Bull. Czech Econometric Society 9 (1999), 57-79
[24] Víšek J. Á.: The least trimmed squares - random carriers. Bull. Czech Econometric Society 10 (1999), 1-30
[25] 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
[26] 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
[27] Víšek J. Á.: Regression with high breakdown point. Robust 2000 (J. Antoch and G. Dohnal, Union of the Czechoslovak Mathematicians and Physicists, Prague 2001, 324-356
[28] 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
[29] Víšek J. Á.: The least weighted squares I. The asymptotic linearity of normal equation. Bull. Czech Econometric Society 9 (2002), 15, 31-58
[30] Víšek J. Á.: The least weighted squares II. Consistency and asymptotic normality. Bull. Czech Econometric Society 9 (2002), 16, 1-28
[31] Víšek J. Á.: Kolmogorov-Smirnov statistics in linear regression. Proc. ROBUST 2006, submitted
[32] Víšek J. Á.: Least trimmed squares - sensitivity study. Proc. Prague Stochastics 2006, submitted
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.