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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

Abstract

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.

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Lucien Birge. Pascal Massart. "Estimation of Integral Functionals of a Density." Ann. Statist. 23 (1) 11 - 29, February, 1995. https://doi.org/10.1214/aos/1176324452

Information

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

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

Subjects:
Primary: 62G05
Secondary: 62G07

Rights: Copyright © 1995 Institute of Mathematical Statistics

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Vol.23 • No. 1 • February, 1995
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