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Tail Behavior of Regression Estimators and Their Breakdown Points

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  • He, Xuming, et al

Abstract

A measure of finite sample estimator performance based on the probability of large deviations is introduced. The tail performance of the least squares estimator is studied in detail, and the authors find that it achieves good tail performance under strictly Gaussian conditions, but performance is extremely poor in the case of heavy-tailed error distributions. Turning to the tail behavior of various robust estimators, they focus on tail performance under heavy (algebraic) tail errors. Perhaps most significantly, it is shown that the authors' finite-sample measure of tail performance is, for heavy tailed error distributions, essentially the same as the finite sample concept of breakdown point introduced by D. L. Donoho and P. J. Huber (1983). Coauthors are Jana Jureckova, Roger Koenker, and Stephen Portnoy. Copyright 1990 by The Econometric Society.

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  • He, Xuming, et al, 1990. "Tail Behavior of Regression Estimators and Their Breakdown Points," Econometrica, Econometric Society, vol. 58(5), pages 1195-1214, September.
    • Handle: RePEc:ecm:emetrp:v:58:y:1990:i:5:p:1195-1214
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    Cited by:

    1. Cizek, P., 2009. "Generalized Methods of Trimmed Moments," Discussion Paper 2009-25, Tilburg University, Center for Economic Research.
    2. Hubert, Mia, 1997. "The breakdown value of the L1 estimator in contingency tables," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 419-425, May.
    3. Giloni, Avi & Simonoff, Jeffrey S. & Sengupta, Bhaskar, 2006. "Robust weighted LAD regression," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3124-3140, July.
    4. Hill, Jonathan B. & Aguilar, Mike, 2013. "Moment condition tests for heavy tailed time series," Journal of Econometrics, Elsevier, vol. 172(2), pages 255-274.
    5. Jurecková, Jana & Koenker, Roger & Portnoy, Stephen, 2001. "Tail behavior of the least-squares estimator," Statistics & Probability Letters, Elsevier, vol. 55(4), pages 377-384, December.
    6. Ke Zhu & Shiqing Ling, 2015. "LADE-Based Inference for ARMA Models With Unspecified and Heavy-Tailed Heteroscedastic Noises," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 784-794, June.
    7. Čížek, Pavel, 2012. "Semiparametric robust estimation of truncated and censored regression models," Journal of Econometrics, Elsevier, vol. 168(2), pages 347-366.
    8. Jurecková, Jana, 2000. "Test of tails based on extreme regression quantiles," Statistics & Probability Letters, Elsevier, vol. 49(1), pages 53-61, August.
    9. Mikosch, Thomas & de Vries, Casper G., 2013. "Heavy tails of OLS," Journal of Econometrics, Elsevier, vol. 172(2), pages 205-221.
    10. Christine Müller, 2011. "Data depth for simple orthogonal regression with application to crack orientation," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(2), pages 135-165, September.
    11. Vijverberg, Wim P. & Hasebe, Takuya, 2015. "GTL Regression: A Linear Model with Skewed and Thick-Tailed Disturbances," IZA Discussion Papers 8898, Institute of Labor Economics (IZA).
    12. Zuo, Yijun, 2003. "Finite sample tail behavior of multivariate location estimators," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 91-105, April.
    13. Mizera, Ivan & Müller, Christine H., 2002. "Breakdown points of Cauchy regression-scale estimators," Statistics & Probability Letters, Elsevier, vol. 57(1), pages 79-89, March.
    14. Neykov, N.M. & Čížek, P. & Filzmoser, P. & Neytchev, P.N., 2012. "The least trimmed quantile regression," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1757-1770.
    15. Jana Jurecková, 2003. "Statistical tests on tail index of a probability distribution," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 151-190.
    16. Gather, Ursula & Einbeck, Jochen & Fried, Roland, 2005. "Weighted Repeated Median Smoothing and Filtering," Technical Reports 2005,33, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    17. Jozef Kušnier & Ivan Mizera, 2001. "Tail Behavior and Breakdown Properties of Equivariant Estimators of Location," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(2), pages 244-261, June.
    18. Salvador Flores, 2015. "Sharp non-asymptotic performance bounds for $$\ell _1$$ ℓ 1 and Huber robust regression estimators," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(4), pages 796-812, December.
    19. Chen, Zhiqiang & E. Tyler, David, 2004. "On the finite sample breakdown points of redescending M-estimates of location," Statistics & Probability Letters, Elsevier, vol. 69(3), pages 233-242, September.
    20. Davies, P. Laurie & Fried, Roland & Gather, Ursula, 2002. "Robust signal extraction for on-line monitoring data," Technical Reports 2002,02, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    21. Jurecková, Jana, 2010. "Finite-sample distribution of regression quantiles," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1940-1946, December.
    22. Tamara Broderick & Ryan Giordano & Rachael Meager, 2020. "An Automatic Finite-Sample Robustness Metric: When Can Dropping a Little Data Make a Big Difference?," Papers 2011.14999, arXiv.org, revised Jul 2023.
    23. Barbe, Ph. & Broniatowski, M., 2004. "Blowing number of a distribution for a statistics and loyal estimators," Statistics & Probability Letters, Elsevier, vol. 69(4), pages 465-475, October.
    24. Bradu, Dan, 1997. "Identification of outliers by means of L1 regression: Safe and unsafe configurations," Computational Statistics & Data Analysis, Elsevier, vol. 24(3), pages 271-281, May.
    25. Avi Giloni & Bhaskar Sengupta & Jeffrey S. Simonoff, 2006. "A mathematical programming approach for improving the robustness of least sum of absolute deviations regression," Naval Research Logistics (NRL), John Wiley & Sons, vol. 53(4), pages 261-271, June.

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