The Annals of Statistics

Smoothing Spline Density Estimation: Theory

Chong Gu and Chunfu Qiu

Full-text: Open access


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

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

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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)

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


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

Export citation