Institute of Mathematical Statistics Collections

Robust error-term-scale estimate

Jan Ámos Víšek

Full-text: Open access

Abstract

A scale-equivariant and regression-invariant estimator of the variance of error terms in the linear regression model is proposed and its consistency proved. The estimator is based on (down)weighting the order statistics of the squared residuals which corresponds to the consistent and scale- and regression-equivariant estimator of the regression coefficients. A small numerical study demonstrating the behaviour of the estimator under the various types of contamination is included.

Chapter information

Source
J. Antoch, M. Hušková and P.K. Sen, eds., Nonparametrics and Robustness in Modern Statistical Inference and Time Series Analysis: A Festschrift in honor of Professor Jana Jurečková (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2010), 254-267

Dates
First available in Project Euclid: 29 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1291044761

Digital Object Identifier
doi:10.1214/10-IMSCOLL725

Subjects
Primary: 62J02: General nonlinear regression
Secondary: 62F35: Robustness and adaptive procedures

Keywords
robustness weighting the order statistics of squared residuals consistency of the scale estimator

Rights
Copyright © 2010, Institute of Mathematical Statistics

Citation

Víšek, Jan Ámos. Robust error-term-scale estimate. Nonparametrics and Robustness in Modern Statistical Inference and Time Series Analysis: A Festschrift in honor of Professor Jana Jurečková, 254--267, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2010. doi:10.1214/10-IMSCOLL725. https://projecteuclid.org/euclid.imsc/1291044761


Export citation

References

  • [1] Agulló, J. (2001). New algorithms for computing the least trimmed squares regression estimators. Computational Statistics and Data Analysis 36 425–439.
  • [2] Benáček, V. and Víšek, J. Á. (2002). Determining factors of trade specialization and growth of a small economy in transition. Impact of the EU opening-up on Czech exports and imports. IIASA, Austria, IR series no. IR-03-001 1–41.
  • [3] Bickel, P. J. (1975). One-step Huber estimates in the linear model. J. Amer. Statist. Assoc. 70 428–433.
  • [4] Breiman, L. (1968). Probability. Addison-Wesley Publishing Company, London.
  • [5] Boček, P. and Lachout, P. (1993). Linear programming approach to LMS-estimation. Memorial volume of Comput. Statist. & Data Analysis 19 129–134.
  • [6] Bramanti, M. C. and Croux, C. (2007). Robust estimators for the fixed effects panel data model. The Econometrics Journal 10 321–540.
  • [7] Chatterjee, S. and Hadi A. S. (1988). Sensitivity Analysis in Linear Regression. J. Wiley & Sons, New York.
  • [8] Croux, C. and Rousseeuw, P. J. (1992). A class of high-breakdown scale estimators based on subranges. Communications in Statistics – Theory and Methods 21 1935 –1951.
  • [9] Čížek, P. and Víšek, J. Á. (2000). The least trimmed squares. User Guide of Explore.
  • [10] Hájek, J. and Šidák, Z. (1967). Theory of Rank Test. Academic Press, New York.
  • [11] Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., and Stahel W. A. (1986). Robust Statistics – The Approach Based on Influence Functions. J. Wiley & Sons, New York.
  • [12] Härdle, W., Hlávka, Z, and Klinke, S. (2000). XploRe Application Guide. Springer Verlag, Heilderberg.
  • [13] Hawkins, D. M. (1994). The feasible solution algorithm for least trimmed squares regression. Computational Statistics and Data Analysis 17, 185–196.
  • [14] Hettmansperger, T. P. and Sheather, S. J. (1992). A cautionary note on the method of Least Median Squares. The American Statistician 46 79–83.
  • [15] Hofmann, M., Gatu, C., and Kontoghiorghes E. J. (2010). An exact least trimmed squares algorithm for a range of coverage values. J. of Computational and Graphical Statistics 19 191–204.
  • [16] Judge, G., Griffiths, W. E., Hill, R. C., Lütkepohl, H., and Lee, T. C. (1982). Introduction to the Theory and Practice of Econometrics. J. Wiley & Sons, New York.
  • [17] Jurečková, J. and Picek J. (2006). Robust Statistical Methods with R. Chapman & Hall, New York.
  • [18] Jurečková, J. and Sen, P. K. (1984). On adaptive scale-equivariant M-estimators in linear models. Statistics and Decisions 2, Suppl. Issue No. 1.
  • [19] Jurečková, J. and Sen, P. K. (1993). Regression rank scores scale statistics and studentization in linear models. Proc. Fifth Prague Symposium on Asymptotic Statistics, Physica Verlag 111–121.
  • [20] Klouda, K. (2007). Algorithms for computing robust regression estimates. Diploma Thesis, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Prague.
  • [21] Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33–50.
  • [22] Marazzi, A. (1992). Algorithms, Routines and S Functions for Robust Statistics. Wadsworth & Brooks/Cole Publishing Company, Belmont.
  • [23] Maronna, R. A. and Yohai, V. J. (1981). The breakdown point of simultaneous general M-estimates of regression and scale. J. of Amer. Statist. Association 86 (415), 699–704.
  • [24] Pison, G., Van Aelst, S., and Willems, G. (2002). Small sample corrections for LTS and MCD. Metrika 55 111–123.
  • [25] Portnoy, S. (1983). 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, D. Martin, eds.), 231–246. Springer-Verlag, New York.
  • [26] Rousseeuw, P. J. (1984). Least median of square regression. J. Amer. Statist. Association 79 871–880.
  • [27] Rousseeuw, P. J. and Driessen, K. (2000). An algorithm for positive-breakdown regression based on concentration steps. In Data Analysis: Scientific Modeling and Practical Application (W. Gaul, O. Opitz, M. Schader, eds.), 335 - 346. Springer-Verlag, Berlin.
  • [28] Rousseeuw, P. J. and Driessen, K. (2002). Fast-LTS in Matlab, code revision 20/04/2006.
  • [29] Rousseeuw, P. J. and Leroy, A. M. (1987). Robust Regression and Outlier Detection. J. Wiley, New York.
  • [30] Stock, J. H. and Trebbi, F. (2003). Who invented instrumental variable regression? Journal of Economic Perspectives 17 177–194.
  • [31] Štěpán, J. (1987). Teorie pravděpodobnosti. Academia, Praha.
  • [32] Víšek, J. Á. (1994). A cautionary note on the method of the Least Median of Squares reconsidered. Trans. Twelfth Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, Academy of Sciences of the Czech Republic, 254–259.
  • [33] Víšek, J. Á. (1996). Sensitivity analysis of M-estimates. Annals of the Institute of Statistical Mathematics 48 469–495.
  • [34] Víšek, J. Á. (1996). On high breakdown point estimation. Computational Statistics 137–146.
  • [35] Víšek, J. Á. (2000). Regression with high breakdown point. Robust 2000 (J. Antoch & G. Dohnal, eds.) Union of the Czech Mathematicians and Physicists, 324–356.
  • [36] Víšek, J. Á. (2002). Sensitivity analysis of M-estimates of nonlinear regression model: Influence of data subsets. Annals of the Institute of Statistical Mathematics 54 261–290.
  • [37] Víšek, J. Á. (2006). The least trimmed squares. Sensitivity study. Proc. Prague Stochastics 2006 (M. Hušková & M. Janžura, eds.), matfyzpress, 728–738.
  • [38] Víšek, J. Á. (2006). Kolmogorov–Smirnov statistics in multiple regression. Proc. ROBUST 2006 (J. Antoch & G. Dohnal, eds.), Union of the Czech Mathematicians and Physicists, 367–374.
  • [39] Víšek, J. Á. (2006). Instrumental weighted variables – algorithm. Proc. COMPSTAT 2006 777–786.
  • [40] Víšek, J. Á. (2009). Consistency of the least weighted squares under heteroscedasticity. Submitted to the Kybernetika.
  • [41] Víšek, J. Á. (2009). Consistency of the instrumental weighted variables. Annals of the Institute of Statistical Mathematics 61 543–578.
  • [42] Víšek, J. Á. (2010). Empirical distribution function under heteroscedasticity. To appear in Statistics.
  • [43] Víšek, J. Á. (2010). Weak sqrtn-consistency of the least weighted squares under heteroscedasticity. Submitted to Acta Universitatis Carolinae – Mathematica et Physica.
  • [44] Wooldridge, J. M. (2001). Econometric Analysis of Cross Section and Panel Data. MIT Press, Cambridge, Massachusetts.
  • [45] Zvára, K. (1989). Regresní analýza (Regression Analysis – in Czech). Academia, Praha.