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Keywords:
robust regression; the least trimmed squares; $\sqrt{n}$-consistency; asymptotic normality
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.
References:
[1] Benáček V. J., Jarolím, M., Víšek J. Á.: Supply-side characteristics and the industrial structure of Czech foreign trade. In: 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 DOI 10.1016/0167-9473(93)E0051-5 | MR 1323269 | Zbl 0875.62292
[3] Breiman L.: Probability. Addison–Wesley Publishing Company, London 1968 MR 0229267 | Zbl 0753.60001
[4] Chatterjee S., Hadi A. S.: Sensitivity Analysis in Linear Regression. Wiley, New York 1988 MR 0939610 | Zbl 0648.62066
[5] Čížek P.: Robust estimation with discrete explanatory variables. In: COMPSTAT 2003, pp. 509–514 MR 1986578
[6] Čížek P., Víšek J. Á.: Least trimmed squares. In: EXPLORE, Application Guide (W. Härdle, Z. Hlavka, and S. Klinke, eds.), Springer–Verlag, Berlin 2000, pp. 49–64
[7] Jurečková J., (1993) P. K. Sen: Regression rank scores scale statistics and studentization in linear models. In: Proc. Fifth Prague Symposium on Asymptotic Statistics, Physica–Verlag, Heidelberg 1993, pp. 111–121 MR 1311932
[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 MR 0829458 | 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 DOI 10.1016/S0167-9473(98)00082-6 | MR 1681450
[10] Hettmansperger T. P., Sheather S. J.: A cautionary note on the method of least median squares. Amer. Statist. 46 (1992), 79–83 MR 1165565
[11] Huber P. J.: Robust Statistics. Wiley, New York 1981 MR 0606374
[12] Liese F., Vajda I.: Consistency of $M$-estimators in general models. J. Multivar. Anal. 50 (1994), 93–114 DOI 10.1006/jmva.1994.1036 | MR 1292610
[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 DOI 10.1007/BF00536192 | MR 0635268
[14] Pollard D.: Asymptotics for least absolute deviation regression estimator. Econometric Theory 7 (1991), 186–199 DOI 10.1017/S0266466600004394 | MR 1128411
[15] Portnoy S.: Tightness of the sequence of empiric c. d.f. processes defined from regression fractiles. In: Robust and Nonlinear Time-Series Analysis (J. Franke, W. Härdle, and D. Martin, eds.), Springer–Verlag, New York 1983, pp. 231–246 MR 0786311
[16] Rousseeuw P. J., Leroy A. M.: Robust Regression and Outlier Detection. Wiley, New York 1987 MR 0914792 | 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 MR 1476149 | Zbl 1167.62388
[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. In: 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 MR 1394545 | Zbl 0933.62015
[21] Víšek J. Á.: Sensitivity analysis $M$-estimates. Ann. Inst. Statist. Math. 48 (1996), 469–495 DOI 10.1007/BF00050849 | MR 1424776
[22] Víšek J. Á.: Diagnostics of regression subsample stability. Probab. Math. Statist. 17 (1997), 2, 231–257 MR 1490803 | 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. In: 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 DOI 10.1016/S0167-9473(99)00068-7 | Zbl 1052.62509
[27] Víšek J. Á.: Regression with high breakdown point. In: Robust 2000 (J. Antoch and G. Dohnal, eds.), 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 DOI 10.1023/A:1022465701229 | MR 1910173 | Zbl 1013.62072
[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. In: Proc. ROBUST 2006, submitted
[32] Víšek J. Á.: Least trimmed squares – sensitivity study. In: Proc. Prague Stochastics 2006, submitted
[33] Zvára K.: Regresní analýza (Regression Analysis – in Czech). Academia, Prague 1989
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