Spatially-adaptive Penalties for Spline Fitting

We study spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. Our 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 GCV. 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, ∗David Ruppert is Professor, School of Operations Research & Industrial Engineering, Cornell University, Ithaca, New York 14853-3801 (E-mail: [email protected]). Ruppert’s research was supported an NSF grant. R.J. Carroll is University Distinguished Professor of Statistics, Nutrition and Toxicology, Texas A&M University, College Station, TX 77843-3143 (E-mail: [email protected]). Carroll’s research was supported by a grant from the National Cancer Institute (CA-57030) and by the Texas A & M Center for Environmental and Rural Health through a grant from the National Institute of Environmental Health Sciences (P30-E509106). Carroll’s work was partially completed during visits to Sonderforschungsbereich 373 at the Humboldt Universität zu Berlin. Matt Wand kindly showed us his manuscript on a comparison of regression spline smoothing method. We thank George Casella, Rob Kass, and Marty Wells for references to the empirical Bayes literature. This paper has benefitted substantially from the constructive comments of two referees and an associate editor.