The Annals of Statistics

Splines as Local Smoothers

Douglas Nychka

Full-text: Open access

Abstract

A smoothing spline is a nonparametric curve estimate that is defined as the solution to a minimization problem. One problem with this representation is that it obscures the fact that a spline, like most other nonparametric estimates, is a local, weighted average of the observed data. This property has been used extensively to study the limiting properties of kernel estimates and it is advantageous to apply similar techniques to spline estimates. Although equivalent kernels have been identified for a smoothing spline, these functions are either not accurate enough for asymptotic approximations or are restricted to equally spaced points. This paper extends this previous work to understand a spline estimate's local properties. It is shown that the absolute value of the spline weight function decreases exponentially away from its center. This result is not asymptotic. The only requirement is that the empirical distribution of the observation points be sufficiently close to a continuous distribution with a strictly positive density function. These bounds are used to derive the asymptotic form for the bias and variance of a first order smoothing spline estimate. The arguments leading to this result can be easily extended to higher order splines.

Article information

Source
Ann. Statist. Volume 23, Number 4 (1995), 1175-1197.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176324704

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aos/1176324704

Mathematical Reviews number (MathSciNet)
MR1353501

Zentralblatt MATH identifier
0842.62025

Subjects
Primary: 62G05: Estimation
Secondary: 62G15: Tolerance and confidence regions

Keywords
Nonparametric regression adaptive smoothing spline equivalent kernel

Citation

Nychka, Douglas. Splines as Local Smoothers. Ann. Statist. 23 (1995), no. 4, 1175--1197. doi:10.1214/aos/1176324704. http://projecteuclid.org/euclid.aos/1176324704.


Export citation