Open Access
Translator Disclaimer
February, 1995 Estimation of Integral Functionals of a Density
Lucien Birge, Pascal Massart
Ann. Statist. 23(1): 11-29 (February, 1995). DOI: 10.1214/aos/1176324452


Let $\varphi$ be a smooth function of $k + 2$ variables. We shall investigate in this paper the rates of convergence of estimators of $T(f) = \int\varphi(f(x), f'(x), \ldots, f^{(k)}(x), x) dx$ when $f$ belongs to some class of densities of smoothness $s$. We prove that, when $s \geq 2k + \frac{1}{4}$, one can define an estimator $\hat{T}_n$ of $T(f)$, based on $n$ i.i.d. observations of density $f$ on the real line, which converges at the semiparametric rate $1/ \sqrt n$. On the other hand, when $s < 2k + \frac{1}{4}, T(f)$ cannot be estimated at a rate faster than $n^{-\gamma}$ with $\gamma = 4(s - k)/\lbrack 4s + 1\rbrack$. We shall also provide some extensions to the multidimensional case. Those results extend previous works of Levit, of Bickel and Ritov and of Donoho and Nussbaum on estimation of quadratic functionals.


Download Citation

Lucien Birge. Pascal Massart. "Estimation of Integral Functionals of a Density." Ann. Statist. 23 (1) 11 - 29, February, 1995.


Published: February, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0848.62022
MathSciNet: MR1331653
Digital Object Identifier: 10.1214/aos/1176324452

Primary: 62G05
Secondary: 62G07

Keywords: Integral functionals , kernel estimators , nonparametric rates of convergence , Quadratic functionals of a density , Semiparametric estimation

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 1 • February, 1995
Back to Top