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Tail Behavior of Regression Estimators and Their Breakdown Points

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  • He, Xuming, et al
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    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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    Bibliographic Info

    Article provided by Econometric Society in its journal Econometrica.

    Volume (Year): 58 (1990)
    Issue (Month): 5 (September)
    Pages: 1195-1214

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    Handle: RePEc:ecm:emetrp:v:58:y:1990:i:5:p:1195-1214

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    Cited by:
    1. Mizera, Ivan & Müller, Christine H., 2002. "Breakdown points of Cauchy regression-scale estimators," Statistics & Probability Letters, Elsevier, vol. 57(1), pages 79-89, March.
    2. 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.
    3. Christine Müller, 2011. "Data depth for simple orthogonal regression with application to crack orientation," Metrika, Springer, vol. 74(2), pages 135-165, September.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Cizek, P., 2008. "Semiparametric Robust Estimation of Truncated and Censored Regression Models," Discussion Paper 2008-34, Tilburg University, Center for Economic Research.
    9. 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.
    10. Cizek, P., 2009. "Generalized Methods of Trimmed Moments," Discussion Paper 2009-25, Tilburg University, Center for Economic Research.
    11. Jozef Kušnier & Ivan Mizera, 2001. "Tail Behavior and Breakdown Properties of Equivariant Estimators of Location," Annals of the Institute of Statistical Mathematics, Springer, vol. 53(2), pages 244-261, June.
    12. Hill, Jonathan B. & Aguilar, Mike, 2013. "Moment condition tests for heavy tailed time series," Journal of Econometrics, Elsevier, vol. 172(2), pages 255-274.
    13. Giloni, Avi & Simonoff, Jeffrey S. & Sengupta, Bhaskar, 2006. "Robust weighted LAD regression," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3124-3140, July.
    14. Jurecková, Jana, 2000. "Test of tails based on extreme regression quantiles," Statistics & Probability Letters, Elsevier, vol. 49(1), pages 53-61, August.
    15. 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.
    16. Jurecková, Jana, 2010. "Finite-sample distribution of regression quantiles," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1940-1946, December.
    17. Zuo, Yijun, 2003. "Finite sample tail behavior of multivariate location estimators," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 91-105, April.
    18. Mikosch, Thomas & de Vries, Casper G., 2013. "Heavy tails of OLS," Journal of Econometrics, Elsevier, vol. 172(2), pages 205-221.

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