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Smoothing Splines and Shape Restrictions

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  • E. Mammen
  • C. Thomas‐Agnan

Abstract

Constrained smoothing splines are discussed under order restrictions on the shape of the function m. We consider shape constraints of the type m(r)≥ 0, i.e. positivity, monotonicity, convexity, .... (Here for an integer r≥ 0, m(r) denotes the rth derivative of m.) The paper contains three results: (1) constrained smoothing splines achieve optimal rates in shape restricted Sobolev classes; (2) they are equivalent to two step procedures of the following type: (a) in a first step the unconstrained smoothing spline is calculated; (b) in a second step the unconstrained smoothing spline is “projected” onto the constrained set. The projection is calculated with respect to a Sobolev‐type norm; this result can be used for two purposes, it may motivate new algorithmic approaches and it helps to understand the form of the estimator and its asymptotic properties; (3) the infinite number of constraints can be replaced by a finite number with only a small loss of accuracy, this is discussed for estimation of a convex function.

Suggested Citation

  • E. Mammen & C. Thomas‐Agnan, 1999. "Smoothing Splines and Shape Restrictions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(2), pages 239-252, June.
  • Handle: RePEc:bla:scjsta:v:26:y:1999:i:2:p:239-252
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    File URL: https://doi.org/10.1111/1467-9469.00147
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