IDEAS home Printed from https://ideas.repec.org/a/spr/alstar/v97y2013i4p387-402.html
   My bibliography  Save this article

On likelihood ratio testing for penalized splines

Author

Listed:
  • Sonja Greven
  • Ciprian Crainiceanu

Abstract

Penalized spline regression using a mixed effects representation is one of the most popular nonparametric regression tools to estimate an unknown regression function $$f(\cdot )$$ . In this context testing for polynomial regression against a general alternative is equivalent to testing for a zero variance component. In this paper, we fill the gap between different published null distributions of the corresponding restricted likelihood ratio test under different assumptions. We show that: (1) the asymptotic scenario is determined by the choice of the penalty and not by the choice of the spline basis or number of knots; (2) non-standard asymptotic results correspond to common penalized spline penalties on derivatives of $$f(\cdot )$$ , which ensure good power properties; and (3) standard asymptotic results correspond to penalized spline penalties on $$f(\cdot )$$ itself, which lead to sizeable power losses under smooth alternatives. We provide simple and easy to use guidelines for the restricted likelihood ratio test in this context. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Sonja Greven & Ciprian Crainiceanu, 2013. "On likelihood ratio testing for penalized splines," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 97(4), pages 387-402, October.
    • Handle: RePEc:spr:alstar:v:97:y:2013:i:4:p:387-402
    DOI: 10.1007/s10182-013-0214-0
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10182-013-0214-0
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10182-013-0214-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    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. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167.
    2. Yingxing Li & David Ruppert, 2008. "On the asymptotics of penalized splines," Biometrika, Biometrika Trust, vol. 95(2), pages 415-436.
    3. Ciprian Crainiceanu & David Ruppert & Gerda Claeskens & M. P. Wand, 2005. "Exact likelihood ratio tests for penalised splines," Biometrika, Biometrika Trust, vol. 92(1), pages 91-103, March.
    4. Gerda Claeskens, 2004. "Restricted likelihood ratio lack‐of‐fit tests using mixed spline models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(4), pages 909-926, November.
    5. Gerda Claeskens & Tatyana Krivobokova & Jean D. Opsomer, 2009. "Asymptotic properties of penalized spline estimators," Biometrika, Biometrika Trust, vol. 96(3), pages 529-544.
    6. Eva Cantoni, 2002. "Degrees-of-freedom tests for smoothing splines," Biometrika, Biometrika Trust, vol. 89(2), pages 251-263, June.
    7. Sonja Greven & Thomas Kneib, 2010. "On the behaviour of marginal and conditional AIC in linear mixed models," Biometrika, Biometrika Trust, vol. 97(4), pages 773-789.
    8. Göran Kauermann & Tatyana Krivobokova & Ludwig Fahrmeir, 2009. "Some asymptotic results on generalized penalized spline smoothing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 487-503, April.
    9. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506.
    10. Ciprian M. Crainiceanu & David Ruppert, 2004. "Likelihood ratio tests in linear mixed models with one variance component," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 165-185, February.
    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. Sonja Greven & Fabian Scheipl, 2016. "Comment," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(516), pages 1568-1573, October.

    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. Chen, Haiqiang & Fang, Ying & Li, Yingxing, 2015. "Estimation And Inference For Varying-Coefficient Models With Nonstationary Regressors Using Penalized Splines," Econometric Theory, Cambridge University Press, vol. 31(4), pages 753-777, August.
    2. repec:wyi:journl:002195 is not listed on IDEAS
    3. repec:hum:wpaper:sfb649dp2013-033 is not listed on IDEAS
    4. Rong Chen & Hua Liang & Jing Wang, 2011. "Determination of linear components in additive models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(2), pages 367-383.
    5. I. Gijbels & I. Prosdocimi & G. Claeskens, 2010. "Nonparametric estimation of mean and dispersion functions in extended generalized linear models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(3), pages 580-608, November.
    6. Huaihou Chen & Yuanjia Wang, 2011. "A Penalized Spline Approach to Functional Mixed Effects Model Analysis," Biometrics, The International Biometric Society, vol. 67(3), pages 861-870, September.
    7. Christian Schellhase & Göran Kauermann, 2012. "Density estimation and comparison with a penalized mixture approach," Computational Statistics, Springer, vol. 27(4), pages 757-777, December.
    8. Simon N. Wood & Zheyuan Li & Gavin Shaddick & Nicole H. Augustin, 2017. "Generalized Additive Models for Gigadata: Modeling the U.K. Black Smoke Network Daily Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1199-1210, July.
    9. Wu, Ximing & Sickles, Robin, 2018. "Semiparametric estimation under shape constraints," Econometrics and Statistics, Elsevier, vol. 6(C), pages 74-89.
    10. repec:wyi:journl:002174 is not listed on IDEAS
    11. Takuma Yoshida, 2016. "Asymptotics and smoothing parameter selection for penalized spline regression with various loss functions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 70(4), pages 278-303, November.
    12. Lee, Wang-Sheng, 2014. "Big and Tall: Is there a Height Premium or Obesity Penalty in the Labor Market?," IZA Discussion Papers 8606, Institute of Labor Economics (IZA).
    13. Kauermann Goeran & Krivobokova Tatyana & Semmler Willi, 2011. "Filtering Time Series with Penalized Splines," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 15(2), pages 1-28, March.
    14. Holland, Ashley D., 2017. "Penalized spline estimation in the partially linear model," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 211-235.
    15. Yuanjia Wang & Huaihou Chen, 2012. "On Testing an Unspecified Function Through a Linear Mixed Effects Model with Multiple Variance Components," Biometrics, The International Biometric Society, vol. 68(4), pages 1113-1125, December.
    16. J. D. Opsomer & G. Claeskens & M. G. Ranalli & G. Kauermann & F. J. Breidt, 2008. "Non‐parametric small area estimation using penalized spline regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 265-286, February.
    17. Michael Wegener & Göran Kauermann, 2017. "Forecasting in nonlinear univariate time series using penalized splines," Statistical Papers, Springer, vol. 58(3), pages 557-576, September.
    18. Lee, Wang-Sheng, 2014. "Is the BMI a Relic of the Past?," IZA Discussion Papers 8637, Institute of Labor Economics (IZA).
    19. Feng, Yuanhua & Härdle, Wolfgang Karl, 2020. "A data-driven P-spline smoother and the P-Spline-GARCH models," IRTG 1792 Discussion Papers 2020-016, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    20. Hulin Wu & Hongqi Xue & Arun Kumar, 2012. "Numerical Discretization-Based Estimation Methods for Ordinary Differential Equation Models via Penalized Spline Smoothing with Applications in Biomedical Research," Biometrics, The International Biometric Society, vol. 68(2), pages 344-352, June.
    21. Takuma Yoshida & Kanta Naito, 2014. "Asymptotics for penalised splines in generalised additive models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(2), pages 269-289, June.
    22. Andrada Ivanescu & Ana-Maria Staicu & Fabian Scheipl & Sonja Greven, 2015. "Penalized function-on-function regression," Computational Statistics, Springer, vol. 30(2), pages 539-568, June.
    23. Peter Pütz & Thomas Kneib, 2018. "A penalized spline estimator for fixed effects panel data models," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 102(2), pages 145-166, April.

    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:spr:alstar:v:97:y:2013:i:4:p:387-402. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.