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Free‐knot polynomial splines with confidence intervals

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  • Wenxin Mao
  • Linda H. Zhao

Abstract

Summary. We construct approximate confidence intervals for a nonparametric regression function, using polynomial splines with free‐knot locations. The number of knots is determined by generalized cross‐validation. The estimates of knot locations and coefficients are obtained through a non‐linear least squares solution that corresponds to the maximum likelihood estimate. Confidence intervals are then constructed based on the asymptotic distribution of the maximum likelihood estimator. Average coverage probabilities and the accuracy of the estimate are examined via simulation. This includes comparisons between our method and some existing methods such as smoothing spline and variable knots selection as well as a Bayesian version of the variable knots method. Simulation results indicate that our method works well for smooth underlying functions and also reasonably well for discontinuous functions. It also performs well for fairly small sample sizes.

Suggested Citation

  • Wenxin Mao & Linda H. Zhao, 2003. "Free‐knot polynomial splines with confidence intervals," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(4), pages 901-919, November.
    • Handle: RePEc:bla:jorssb:v:65:y:2003:i:4:p:901-919
    DOI: 10.1046/j.1369-7412.2003.00422.x
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    Cited by:

    1. Laura M. Sangalli & Piercesare Secchi & Simone Vantini & Alessandro Veneziani, 2009. "Efficient estimation of three‐dimensional curves and their derivatives by free‐knot regression splines, applied to the analysis of inner carotid artery centrelines," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 58(3), pages 285-306, July.
    2. Lawrence Brown & Xin Fu & Linda Zhao, 2011. "Confidence intervals for nonparametric regression," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(1), pages 149-163.
    3. Binder, Harald & Sauerbrei, Willi, 2008. "Increasing the usefulness of additive spline models by knot removal," Computational Statistics & Data Analysis, Elsevier, vol. 52(12), pages 5305-5318, August.
    4. Holger Dette & Viatcheslav Melas & Andrey Pepelyshev, 2011. "Optimal design for smoothing splines," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(5), pages 981-1003, October.
    5. Dette, Holger & Melas, Viatcheslav B. & Pepelyshev, Andrey, 2006. "Optimal designs for free knot least squares splines," Technical Reports 2006,34, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    6. Nielsen, J.D. & Dean, C.B., 2008. "Adaptive functional mixed NHPP models for the analysis of recurrent event panel data," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3670-3685, March.
    7. Na Li & Xingzhong Xu & Xuhua Liu, 2011. "Testing the constancy in varying-coefficient regression models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(3), pages 409-438, November.
    8. Göran Kauermann & Timo Teuber & Peter Flaschel, 2012. "Exploring US Business Cycles with Bivariate Loops Using Penalized Spline Regression," Computational Economics, Springer;Society for Computational Economics, vol. 39(4), pages 409-427, April.
    9. J. D. Nielsen & C. B. Dean, 2008. "Clustered Mixed Nonhomogeneous Poisson Process Spline Models for the Analysis of Recurrent Event Panel Data," Biometrics, The International Biometric Society, vol. 64(3), pages 751-761, September.

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