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Spatially-adaptive Penalties for Spline Fitting

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The paper studies spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. The estimates are pth degree piecewise polynomials with p - 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty on the jumps of the pth derivative at the knots. To be spatially adaptive, the logarithm of the penalty is itself a linear spline but with relatively few knots and with values at the knots chosen to minimize the generalized cross validation (GCV) criterion. This locally-adaptive spline estimator is compared with other spline estimators in the literature such as cubic smoothing splines and knot-selection techniques for least squares regression. Our estimator can be interpreted as an empirical Bayes estimate for a prior allowing spatial heterogeneity. In cases of spatially heterogeneous regression functions, empirical Bayes confidence intervals using this prior achieve better pointwise coverage probabilities than confidence intervals based on a global-penalty parameter. The method is developed first for univariate models and then extended to additive models.
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Keywords: Bayesian inference; additive models; confidence intervals; hierarchical Bayesian model; regression splines.

Document Type: Research Article

Affiliations: 1: School of Operations Research & Industrial Engineering, Cornell University, Ithaca, New York, USA 2: Statistics, Nutrition and Toxicology, Texas A & M University, TX 77843-3143, USA

Publication date: June 1, 2000

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