Derivation of equivalent kernel for general spline smoothing: A systematic approach

Felix Abramovich*, Vadim Grinshtein

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We consider first the spline smoothing nonparametric estimation with variable smoothing parameter and arbitrary design density function and show that the corresponding equivalent kernel can be approximated by the Green function of a certain linear differential operator. Furthermore, we propose to use the standard (in applied mathematics and engineering) method for asymptotic solution of linear differential equations, known as the Wentzel-Kramers-Brillouin method, for systematic derivation of an asymptotically equivalent kernel in this general case. The corresponding results for polynomial splines are a special case of the general solution. Then, we show how these ideas can be directly extended to the very general L-spline smoothing.

Original languageEnglish
Pages (from-to)359-379
Number of pages21
Issue number2
StatePublished - 1999


  • Green's function
  • L-smoothing spline
  • Nonparametric regression
  • Variable smoothing parameter
  • Wentzel-Kramers-Brillouin method


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