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Locally adaptive regression splines. (English) Zbl 0871.62040
Summary: Least squares penalized regression estimates with total variation penalties are considered. It is shown that these estimators are least squares splines with locally data adaptive placed knot points. The definition of these variable knot splines as minimizers of global functionals can be used to study their asymptotic properties. In particular, these results imply that the estimates adapt well to spatially inhomogeneous smoothness. We show rates of convergence in bounded variation function classes and discuss pointwise limiting distributions. An iterative algorithm based on stepwise addition and deletion of knot points is proposed and its consistency proved.

MSC:
62G07 Density estimation
62G30 Order statistics; empirical distribution functions
62G20 Asymptotic properties of nonparametric inference
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