IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v100y2009i4p622-635.html
   My bibliography  Save this article

Distribution-free tests for polynomial regression based on simplicial depth

Author

Listed:
  • Wellmann, Robin
  • Harmand, Peter
  • Müller, Christine H.

Abstract

A general approach for developing distribution-free tests for general linear models based on simplicial depth is presented. In most relevant cases, the test statistic is a degenerated U-statistic so that the spectral decomposition of the conditional expectation of the kernel function is needed to derive the asymptotic distribution. A general formula for this conditional expectation is derived. Then it is shown how this general formula can be specified for polynomial regression. Based on the specified form, the spectral decomposition and thus the asymptotic distribution is derived for polynomial regression of arbitrary degree. The power of the new test is compared via simulation with other tests. An application on cubic regression demonstrates the applicability of the new tests and in particular their outlier robustness.

Suggested Citation

  • Wellmann, Robin & Harmand, Peter & Müller, Christine H., 2009. "Distribution-free tests for polynomial regression based on simplicial depth," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 622-635, April.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:4:p:622-635
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(08)00161-9
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Ivan Mizera & Christine H. Muller, 2004. "Location-Scale Depth," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 949-966, December.
    2. Van Aelst, Stefan & Rousseeuw, Peter J. & Hubert, Mia & Struyf, Anja, 2002. "The Deepest Regression Method," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 138-166, April.
    3. Wellmann, R. & Katina, S. & Muller, Ch.H., 2007. "Calculation of simplicial depth estimators for polynomial regression with applications," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 5025-5040, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Christoph Kustosz & Christine Müller, 2014. "Analysis of crack growth with robust, distribution-free estimators and tests for non-stationary autoregressive processes," Statistical Papers, Springer, vol. 55(1), pages 125-140, February.
    2. Wellmann, Robin & Müller, Christine H., 2010. "Tests for multiple regression based on simplicial depth," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 824-838, April.
    3. Wellmann, Robin & Müller, Christine H., 2010. "Depth notions for orthogonal regression," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2358-2371, November.
    4. Liesa Denecke & Christine Müller, 2014. "New robust tests for the parameters of the Weibull distribution for complete and censored data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(5), pages 585-607, July.
    5. Christoph P. Kustosz & Anne Leucht & Christine H. MÜller, 2016. "Tests Based on Simplicial Depth for AR(1) Models With Explosion," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(6), pages 763-784, November.
    6. Kevin Leckey & Dennis Malcherczyk & Melanie Horn & Christine H. Müller, 2023. "Simple powerful robust tests based on sign depth," Statistical Papers, Springer, vol. 64(3), pages 857-882, June.
    7. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wellmann, Robin & Müller, Christine H., 2010. "Depth notions for orthogonal regression," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2358-2371, November.
    2. Wellmann, Robin & Müller, Christine H., 2010. "Tests for multiple regression based on simplicial depth," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 824-838, April.
    3. 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.
    4. Ursula Gather & Karen Schettlinger & Roland Fried, 2006. "Online signal extraction by robust linear regression," Computational Statistics, Springer, vol. 21(1), pages 33-51, March.
    5. Zuo, Yijun, 2021. "Computation of projection regression depth and its induced median," Computational Statistics & Data Analysis, Elsevier, vol. 158(C).
    6. 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.
    7. Christmann, Andreas & Steinwart, Ingo & Hubert, Mia, 2006. "Robust Learning from Bites for Data Mining," Technical Reports 2006,03, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    8. Wellmann, R. & Katina, S. & Muller, Ch.H., 2007. "Calculation of simplicial depth estimators for polynomial regression with applications," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 5025-5040, June.
    9. Denecke, Liesa & Müller, Christine H., 2011. "Robust estimators and tests for bivariate copulas based on likelihood depth," Computational Statistics & Data Analysis, Elsevier, vol. 55(9), pages 2724-2738, September.
    10. Christmann, Andreas & Steinwart, Ingo & Hubert, Mia, 2007. "Robust learning from bites for data mining," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 347-361, September.
    11. Yijun Zuo, 2020. "Depth Induced Regression Medians and Uniqueness," Stats, MDPI, vol. 3(2), pages 1-13, April.
    12. Chernozhukov, Victor & Hong, Han, 2003. "An MCMC approach to classical estimation," Journal of Econometrics, Elsevier, vol. 115(2), pages 293-346, August.
    13. Stephan Morgenthaler, 2007. "A survey of robust statistics," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 15(3), pages 271-293, February.
    14. Zuo, Yijun, 2024. "Non-asymptotic robustness analysis of regression depth median," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    15. Debruyne, M. & Hubert, M. & Portnoy, S. & Vanden Branden, K., 2008. "Censored depth quantiles," Computational Statistics & Data Analysis, Elsevier, vol. 52(3), pages 1604-1614, January.
    16. Ryan Cumings-Menon, 2022. "Differentially Private Estimation via Statistical Depth," Papers 2207.12602, arXiv.org.
    17. Stephan Morgenthaler, 2007. "A survey of robust statistics," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 15(3), pages 271-293, February.
    18. Müller, Christine H., 2005. "Depth estimators and tests based on the likelihood principle with application to regression," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 153-181, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:100:y:2009:i:4:p:622-635. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.