Previous |  Up |  Next

Article

Keywords:
robust regression; the least trimmed squares; consistency; discussion of assumptions and of algorithm for evaluation of estimator
Summary:
The consistency of the least trimmed squares estimator (see Rousseeuw [Rous] or Hampel et al. [HamRonRouSta]) is proved under general conditions. The assumptions employed in paper are discussed in details to clarify the consequences for the applications.
References:
[1] Andrews D. W. K.: Consistency in nonlinear econometric models: A generic uniform law of large numbers. Econometrica 55 (1987), 1465–1471 DOI 10.2307/1913568 | MR 0923471 | Zbl 0646.62101
[2] Bickel P. J.: One-step Huber estimates in the linear model. J. Amer. Statist. Assoc. 70 (1975), 428–433 DOI 10.1080/01621459.1975.10479884 | MR 0386168 | Zbl 0322.62038
[3] Breiman L.: Probability. Addison–Wesley, 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] Csőrgő M., Révész P.: Strong Approximation in Probability and Statistics. Akademia Kiadó, Budapest 1981 MR 0666546
[6] Dhrymes P. J.: Introductory Econometrics. Springer–Verlag, New York 1978 MR 0545505 | Zbl 0388.62096
[7] Jurečková J., Sen P. K.: 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] Hettmansperger T. P., Sheather S. J.: A cautionary note on the method of least median squares. Amer. Statist. 46 (1992), 79–83 MR 1165565
[10] 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
[11] 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
[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 MR 0102205
[14] Rousseeuw P. J.: Least median of square regression. J. Amer. Statist. Assoc. 79 (1984), 871–880 DOI 10.1080/01621459.1984.10477105 | MR 0770281
[15] Rousseeuw P. J., Leroy A. M.: Robust Regression and Outlier Detection. Wiley, New York 1987 MR 0914792 | 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 MR 1420128 | 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 MR 1476149 | 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. In: Proc. Computational Statistics, COMPSTAT 2004 (J. Antoch, ed.), Physica–Verlag, Springer 2004, pp. 539–555 MR 2173049
[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.: Ekonometrie I (Econometrics I). Carolinum, Publishing House of Charles University, Prague 1997
[23] Víšek J. Á.: Robust specification test. In: Proc. Prague Stochastics’98 (M. Hušková, P. Lachout, and J. Á. Víšek, eds.), Union of Czechoslovak Mathematicians and Physicists, Prague 1998, pp. 581–586
[24] Víšek J. Á.: Robust instruments. In: Robust’98 (J. Antoch and G. Dohnal, eds.), 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. In: 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 DOI 10.1016/S0167-9473(99)00068-7 | Zbl 1052.62509
[29] Víšek J. Á.: Regression with high breakdown point. In: Robust 2000 (J. Antoch and G. Dohnal, eds.), Union of Czechoslovak Mathematicians and Physicists, Prague 2001, pp. 324–356
[30] Víšek J. Á.: A new paradigm of point estimation. In: Proc. Data Analysis 2000/II, Modern Statistical Methods – Modelling, Regression, Classification and Data Mining (K. Kupka, ed.), 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 DOI 10.1023/A:1022465701229 | MR 1910173 | Zbl 1013.62072
[32] Víšek J. Á.: $\sqrt{n}$-consistency of empirical distribution function of residuals in linear regression model. Probab. Lett., submitted
[33] Zvára K.: Regresní analýza (Regression Analysis). Academia, Prague 1989
Partner of
EuDML logo