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Semiparametric regression with shape-constrained penalized splines

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  • Hazelton, Martin L.
  • Turlach, Berwin A.
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    Abstract

    In semiparametric regression models, penalized splines can be used to describe complex, non-linear relationships between the mean response and covariates. In some applications it is desirable to restrict the shape of the splines so as to enforce properties such as monotonicity or convexity on regression functions. We describe a method for imposing such shape constraints on penalized splines within a linear mixed model framework. We employ Markov chain Monte Carlo (MCMC) methods for model fitting, using a truncated prior distribution to impose the requisite shape restrictions. We develop a computationally efficient MCMC sampler by using a correspondingly truncated multivariate normal proposal distribution, which is a restricted version of the approximate sampling distribution of the model parameters in an unconstrained version of the model. We also describe a cheap approximation to this methodology that can be applied for shape-constrained scatterplot smoothing. Our methods are illustrated through two applications, the first involving the length of dugongs and the second concerned with growth curves for sitka spruce trees.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0167947311001472
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    Bibliographic Info

    Article provided by Elsevier in its journal Computational Statistics & Data Analysis.

    Volume (Year): 55 (2011)
    Issue (Month): 10 (October)
    Pages: 2871-2879

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    Handle: RePEc:eee:csdana:v:55:y:2011:i:10:p:2871-2879

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    Web page: http://www.elsevier.com/locate/csda

    Related research

    Keywords: Linear mixed model MCMC Shape constraint Spline Truncated multivariate normal;

    References

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    1. E. Mammen, 1999. "Smoothing Splines and Shape Restrictions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 26(2), pages 239-252.
    2. Matzkin, Rosa L, 1991. "Semiparametric Estimation of Monotone and Concave Utility Functions for Polychotomous Choice Models," Econometrica, Econometric Society, vol. 59(5), pages 1315-27, September.
    3. Marx, Brian D. & Eilers, Paul H. C., 1998. "Direct generalized additive modeling with penalized likelihood," Computational Statistics & Data Analysis, Elsevier, vol. 28(2), pages 193-209, August.
    4. Brezger, Andreas & Lang, Stefan, 2006. "Generalized structured additive regression based on Bayesian P-splines," Computational Statistics & Data Analysis, Elsevier, vol. 50(4), pages 967-991, February.
    5. Thomas S. Shively & Thomas W. Sager & Stephen G. Walker, 2009. "A Bayesian approach to non-parametric monotone function estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 159-175.
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