Spline adaptation to smoothness

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This article is devoted to spline adaptation to unknown degree and type of inhomogeneous smoothness over large function classes in d-dimensional nonparametric regression. In particular, we study (1) the selection bias introduced by using the unbiased risk estimator for knot selection and (2) convergence properties of an adaptive regression spline (with variable multiple knots) estimator. The selection bias is shown to be of much smaller order of the ideal loss (risk), which, in turn, leads to the optimal rate of convergence of the adaptive spline estimator over large Besov classes. This implies that the regression spline estimator indeed shares optimal properties with the wavelet shrinkage and local kernel estimators.

Original languageEnglish (US)
Pages (from-to)391-409
Number of pages19
JournalJournal of Statistical Planning and Inference
Issue number1-2
StatePublished - Nov 1 2002

Bibliographical note

Funding Information:
Supported in part by a NSA Grant MDA904-98-1-0030.

Copyright 2008 Elsevier B.V., All rights reserved.


  • Besov classes
  • Discontinuity
  • Inhomogeneous
  • Nonparametric regression
  • Smoothness of unknown degree and type
  • Unbiased risk estimator


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