The Annals of Statistics

Spline Smoothing: The Equivalent Variable Kernel Method

B. W. Silverman

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The spline smoothing approach to nonparametric regression and curve estimation is considered. It is shown that, in a certain sense, spline smoothing corresponds approximately to smoothing by a kernel method with bandwidth depending on the local density of design points. Some exact calculations demonstrate that the approximation is extremely close in practice. Consideration of kernel smoothing methods demonstrates that the way in which the effective local bandwidth behaves in spline smoothing has desirable properties. Finally, the main result of the paper is applied to the related topic of penalized maximum likelihood probability density estimates; a heuristic discussion shows that these estimates should adapt well in the tails of the distribution.

Article information

Ann. Statist., Volume 12, Number 3 (1984), 898-916.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62J05: Linear regression 65D10: Smoothing, curve fitting 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Nonparametric regression variable kernel splines roughness penalty weight function adaptive smoothing Sobolev space penalized maximum likelihood curve estimation density estimation


Silverman, B. W. Spline Smoothing: The Equivalent Variable Kernel Method. Ann. Statist. 12 (1984), no. 3, 898--916. doi:10.1214/aos/1176346710.

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