Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T04:59:14.890Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERAL TRIMMED ESTIMATION: ROBUST APPROACH TO NONLINEAR AND LIMITED DEPENDENT VARIABLE MODELS

Published online by Cambridge University Press:  09 July 2008

Pavel Čížek
Pavel Čížek
Affiliation:
Tilburg University
Get access Rights & Permissions [Opens in a new window]

Abstract

High-breakdown-point regression estimators protect against large errors and data contamination. We generalize the concept of trimming used by many of these robust estimators, such as the least trimmed squares and maximum trimmed likelihood, and propose a general trimmed estimator, which renders robust estimators applicable far beyond the standard (non)linear regression models. We derive here the consistency and asymptotic distribution of the proposed general trimmed estimator under mild β-mixing conditions and demonstrate its applicability in nonlinear regression and limited dependent variable models.

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 6 , December 2008 , pp. 1500 - 1529
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Agulló, J. (2001) New algorithms for computing the least trimmed squares regression estimator. Computational Statistics & Data Analysis 36, 425–439.Google Scholar
Albert, A. & Anderson, J.A. (1984) On the existence of maximum likelihood estimates in logistic regression models. Biometrika 71, 1–10.Google Scholar
Andrews, D.W.K. (1988) Laws of large numbers for dependent non-identically distributed random variables. Econometric Theory 4, 458–467.CrossRefGoogle Scholar
Andrews, D.W.K. (1992) Generic uniform convergence. Econometric Theory 8, 241–257.Google Scholar
Andrews, D.W.K. (1993) An introduction to econometric applications of empirical process theory for dependent random variables. Econometric Reviews 12, 183–216.Google Scholar
Arcones, M.A. & Yu, B. (1994) Central limit theorems for empirical and U-processes of stationary mixing sequences. Journal of Theoretical Probability 7, 47–71.CrossRefGoogle Scholar
Bassett, G.W. (1991) Equivariant, monotonic, 50\% breakdown estimators. American Statistician 45, 135–137.Google Scholar
Čížek, P. (2002) Robust estimation with discrete explanatory variables. In Härdle, W. & Rönz, B. (eds.), COMPSTAT 2002—Proceedings in Computational Statistics, pp. 509–514. Springer-Verlag.Google Scholar
Čížek, P. (2004a) Smoothed local L-estimation with an application. In Hubert, M., Pison, G., Struyf, A., & Aelst, S. (eds.), Theory and Applications of Recent Robust Methods, pp. 59–70. Birhaüser.Google Scholar
Čížek, P. (2004b) Asymptotics of Least Trimmed Squares Regression. CentER Discussion paper 72/2004, Tilburg University.CrossRefGoogle Scholar
Čížek, P. (2006a) Least trimmed squares in nonlinear regression under dependence. Journal of Statistical Planning and Inference 136, 3967–3988.CrossRefGoogle Scholar
Čížek, P. (2006b) Trimmed likelihood-based estimation in binary regression models. Austrian Journal of Statistics 35, 223–232.Google Scholar
Čížek, P. (2007) General Trimmed Estimation: Robust Approach to Nonlinear and Limited Dependent Variable Models. CentER Discussion paper 65/2007, Tilburg University.CrossRefGoogle Scholar
Čížek, P. & Víšek, J.Á. (2000) Least trimmed squares. In Härdle, W., Hlávka, Z., & Klinke, S. (eds.), XploRe Application Guide, pp. 49–64. Springer-Verlag.CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.Google Scholar
Doukhan, P., Massart, P., & Rio, E. (1995) Invariance principles for absolutely regular empirical processes. Annales de l'Institut Henri Poincaré 31, 393–427.Google Scholar
Genton, M.G. & Lucas, A. (2003) Comprehensive definitions of breakdown points for independent and dependent observations. Journal of the Royal Statistical Society, Series B 65, 81–94.Google Scholar
Gerfin, M. (1996) Parametric and semi-parametric estimation of the binary response model of labour market participation. Journal of Applied Econometrics 11, 321–339.Google Scholar
Gervini, D. & Yohai, V.J. (2002) A class of robust and fully efficient regression estimators. Annals of Statistics 30, 583–616.CrossRefGoogle Scholar
Gilloni, A. & Padberg, M. (2002) Least trimmed squares regression, least median squares regression, and mathematical programming. Mathematical and Computer Modelling 35, 1043–1060.Google Scholar
Gourieroux, C. & Monfort, A. (1981) Asymptotic properties of the maximum likelihood estimator in dichotomous logit models. Journal of Econometrics 17, 83–97.Google Scholar
Hadi, A. & Luceno, A. (1997) Maximum trimmed likelihood estimators: A unified approach, examples and algorithms. Computational Statistics & Data Analysis 25, 251–272.CrossRefGoogle Scholar
Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., & Stahel, W.A. (1986) Robust Statistics, the Approach Based on Influence Function. Wiley.Google Scholar
Hawkins, D.M. & Olive, D. (1999) Applications and algorithms for least trimmed sum of absolute deviations regression. Computational Statistics & Data Analysis 32, 119–134.Google Scholar
Hössjer, O. (1994) Rank-based estimates in the linear model with high breakdown point. Journal of the American Statistical Association 89, 149–158.Google Scholar
Kelly, M. (1997) Do noise traders influence stock prices? Journal of Money, Credit and Banking 29, 351–363.Google Scholar
Knez, P.J. & Ready, M.J. (1997) On the robustness of size and book-to-market in cross-sectional regressions. Journal of Finance 52, 1355–1382.Google Scholar
Manski, C.F. (1988) Analog Estimation Methods in Econometrics. Chapman and Hall.Google Scholar
Müller, C.H. & Neykov, N.K. (2003) Breakdown points of trimmed likelihood estimators and related estimators in generalized linear models. Journal of Statistical Planning and Inference 116, 503–519.Google Scholar
Neykov, N.M. & Neytchev, P.N. (1990) A robust alternative of the maximum likelihood estimator. Short Communications of COMPSTAT, Dubrovnik 1990, 99–100.Google Scholar
Pakes, A. & Pollard, D. (1989) Simulation and the asymptotics of optimization estimators. Econometrica 57, 1027–1057.CrossRefGoogle Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer-Verlag.CrossRefGoogle Scholar
Powell, J.L. (1986) Symmetrically trimmed least squares estimation for Tobit models. Econometrica 54, 1435–1460.Google Scholar
Rousseeuw, P.J. (1984) Least median of squares regression. Journal of the American Statistical Association 79, 871–880.Google Scholar
Rousseeuw, P.J. (1985) Multivariate estimation with high breakdown point. In Grossman, W., Pflug, G., Vincze, I., & Wertz, W. (eds.), Mathematical Statistics and Applications, vol. B, pp. 283–297. Reidel.Google Scholar
Rousseeuw, P.J. (1997) Introduction to positive-breakdown methods. In Maddala, G.S. & Rao, C.R. (eds.), Handbook of Statistics, vol. 15: Robust Inference. pp. 101–121. Elsevier.Google Scholar
Rousseeuw, P.J. & Leroy, A.M. (2003) Robust Regression and Outlier Detection. Wiley.Google Scholar
Sakata, S. & White, H. (1998) High breakdown point conditional dispersion estimation with application to S&P 500 daily returns volatility. Econometrica 66, 529–567.Google Scholar
Salibian-Barrera, M. & Zamar, R.H. (2002) Bootstrapping robust estimates of regression. Annals of Statistics 30, 556–582.Google Scholar
Silvapulle, M.J. (1981) On the existence of maximum likelihood estimates for the binomial response models. Journal of the Royal Statistical Society Series B 43, 310–313.Google Scholar
Stromberg, A.J. (1993) High breakdown estimation of nonlinear regression parameters. Journal of the American Statistical Association 88, 237–244.Google Scholar
Stromberg, A.J. & Ruppert, D. (1992) Breakdown in nonlinear regression. Journal of the American Statistical Association 87, 991–997.Google Scholar
Tableman, M. (1994) The asymptotics of the least trimmed absolute deviations (LTAD) estimator. Statistics and Probability Letters 19, 329–337.CrossRefGoogle Scholar
Temple, J.R.W. (1998) Robustness tests of the augmented Solow model. Journal of Applied Econometrics 13, 361–375.3.0.CO;2-1>CrossRefGoogle Scholar
Van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes: With Applications to Statistics. Springer-Verlag.Google Scholar
Vandev, D.L. & Neykov, N.M. (1993) Robust maximum likelihood in the Gaussian case. In Morgenthaler, S., Ronchetti, E., & Stahel, W.A. (eds.), New Directions in Data Analysis and Robustness, pp. 259–264. Birkhäuser.Google Scholar
Vandev, D.L. & Neykov, N.M. (1998) About regression estimators with high breakdown point. Statistics 32, 111–129.CrossRefGoogle Scholar
Víšek, J.Á. (2000) On the diversity of estimates. Computational Statistics & Data Analysis 34, 67–89.Google Scholar
Víšek, J.Á. (2002) The least weighted squares II. Consistency and asymptotic normality. Bulletin of the Czech Econometric Society 16, 1–28.Google Scholar
Willems, G. & van Aelst, S. (2005) Fast and robust bootstrap for LTS. Computational Statistics & Data Analysis 48, 703–715.Google Scholar
Zaman, A., Rousseeuw, P.J., & Orhan, M. (2001) Econometric applications of high-breakdown robust regression techniques. Economics Letters 71, 1–8.Google Scholar
Zinde-Walsh, V. (2002) Asymptotic theory for some high breakdown point estimators. Econometric Theory 18, 1172–1196.Google Scholar