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Smoothing parameter selection for a class of semiparametric linear models

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  • Philip T. Reiss
  • R. Todd Ogden

Abstract

Spline-based approaches to non-parametric and semiparametric regression, as well as to regression of scalar outcomes on functional predictors, entail choosing a parameter controlling the extent to which roughness of the fitted function is penalized. We demonstrate that the equations determining two popular methods for smoothing parameter selection, generalized cross-validation and restricted maximum likelihood, share a similar form that allows us to prove several results which are common to both, and to derive a condition under which they yield identical values. These ideas are illustrated by application of functional principal component regression, a method for regressing scalars on functions, to two chemometric data sets. Copyright (c) 2009 Royal Statistical Society.

Suggested Citation

  • Philip T. Reiss & R. Todd Ogden, 2009. "Smoothing parameter selection for a class of semiparametric linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 505-523.
  • Handle: RePEc:bla:jorssb:v:71:y:2009:i:2:p:505-523
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    References listed on IDEAS

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    1. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167, March.
    2. Eva Cantoni, 2002. "Degrees-of-freedom tests for smoothing splines," Biometrika, Biometrika Trust, vol. 89(2), pages 251-263, June.
    3. Reiss, Philip T. & Ogden, R. Todd, 2007. "Functional Principal Component Regression and Functional Partial Least Squares," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 984-996, September.
    4. Cardot, Hervé, 2002. "Spatially Adaptive Splines for Statistical Linear Inverse Problems," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 100-119, April.
    5. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506, March.
    6. 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.
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    Citations

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    Cited by:

    1. Reiss Philip T. & Huang Lei & Mennes Maarten, 2010. "Fast Function-on-Scalar Regression with Penalized Basis Expansions," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-30, August.
    2. Tatyana Krivobokova, 2011. "Smoothing parameter selection in two frameworks for penalized splines," Courant Research Centre: Poverty, Equity and Growth - Discussion Papers 85, Courant Research Centre PEG, revised 18 Oct 2012.
    3. Christian Schellhase & Göran Kauermann, 2012. "Density estimation and comparison with a penalized mixture approach," Computational Statistics, Springer, vol. 27(4), pages 757-777, December.
    4. Manuel Wiesenfarth & Carlos Matías Hisgen & Thomas Kneib & Carmen Cadarso-Suarez, 2014. "Bayesian Nonparametric Instrumental Variables Regression Based on Penalized Splines and Dirichlet Process Mixtures," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(3), pages 468-482, July.
    5. Marra, Giampiero & Wood, Simon N., 2011. "Practical variable selection for generalized additive models," Computational Statistics & Data Analysis, Elsevier, vol. 55(7), pages 2372-2387, July.
    6. Lee, Dae-Jin & Durbán, María & Eilers, Paul, 2013. "Efficient two-dimensional smoothing with P-spline ANOVA mixed models and nested bases," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 22-37.
    7. Marra, Giampiero & Radice, Rosalba, 2013. "Estimation of a regression spline sample selection model," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 158-173.
    8. Tatyana Krivobokova, 2013. "Smoothing parameter selection in two frameworks for penalized splines," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 725-741, September.
    9. 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.
    10. repec:rsr:journl:v:65:y:2017:i:2:p:81-104 is not listed on IDEAS
    11. Usset, Joseph & Staicu, Ana-Maria & Maity, Arnab, 2016. "Interaction models for functional regression," Computational Statistics & Data Analysis, Elsevier, vol. 94(C), pages 317-329.
    12. repec:bla:istatr:v:85:y:2017:i:2:p:228-249 is not listed on IDEAS
    13. Simon N. Wood, 2011. "Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(1), pages 3-36, January.
    14. 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.
    15. Silas Bergen & Lianne Sheppard & Joel D. Kaufman & Adam A. Szpiro, 2016. "Multipollutant measurement error in air pollution epidemiology studies arising from predicting exposures with penalized regression splines," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 65(5), pages 731-753, November.

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