The Annals of Statistics

Spline Smoothing: The Equivalent Variable Kernel Method

B. W. Silverman

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176346710

Digital Object Identifier
doi:10.1214/aos/1176346710

Mathematical Reviews number (MathSciNet)
MR751281

Zentralblatt MATH identifier
0547.62024

JSTOR
links.jstor.org

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

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

Citation

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


Export citation