The Annals of Statistics
- Ann. Statist.
- Volume 21, Number 1 (1993), 217-234.
Smoothing Spline Density Estimation: Theory
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.
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)
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