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Theory & Methods: Spatially‐adaptive Penalties for Spline Fitting
Author(s) -
Ruppert David,
Carroll Raymond J.
Publication year - 2000
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/1467-842x.00119
Subject(s) - mathematics , smoothing spline , estimator , pointwise , spline (mechanical) , smoothing , piecewise , univariate , nonparametric regression , thin plate spline , mathematical optimization , statistics , spline interpolation , mathematical analysis , bilinear interpolation , multivariate statistics , structural engineering , engineering
The paper studies spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. The estimates are p th degree piecewise polynomials withp − 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 thep th 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.