The Annals of Statistics

Smoothing Spline Density Estimation: Theory

Chong Gu and Chunfu Qiu

Full-text: Open access

Abstract

In this article, a class of penalized likelihood probability density estimators is proposed and studied. The true log density is assumed to be a member of a reproducing kernel Hilbert space on a finite domain, not necessarily univariate, and the estimator is defined as the unique unconstrained minimizer of a penalized log likelihood functional in such a space. Under mild conditions, the existence of the estimator and the rate of convergence of the estimator in terms of the symmetrized Kullback-Leibler distance are established. To make the procedure applicable, a semiparametric approximation of the estimator is presented, which sits in an adaptive finite dimensional function space and hence can be computed in principle. The theory is developed in a generic setup and the proofs are largely elementary. Algorithms are yet to follow.

Article information

Source
Ann. Statist., Volume 21, Number 1 (1993), 217-234.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349023

Digital Object Identifier
doi:10.1214/aos/1176349023

Mathematical Reviews number (MathSciNet)
MR1212174

Zentralblatt MATH identifier
0770.62030

JSTOR
links.jstor.org

Subjects
Primary: 62G07: Density estimation
Secondary: 65D07: Splines 65D10: Smoothing, curve fitting 41A25: Rate of convergence, degree of approximation 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

Keywords
Density estimation penalized likelihood rate of convergence reproducing kernel Hilbert space semiparametric approximation smoothing splines

Citation

Gu, Chong; Qiu, Chunfu. Smoothing Spline Density Estimation: Theory. Ann. Statist. 21 (1993), no. 1, 217--234. doi:10.1214/aos/1176349023. https://projecteuclid.org/euclid.aos/1176349023


Export citation