Z.Consider the problem of estimating $\int \Phi(f)$, where $\Phi$ is a smooth function and f is a density with given order of regularity s. Special attention is paid to the case $\Phi(t) = t^3$. It has been shown that for low values of s the $n^{-1/2}$ rate of convergence is not achievable uniformly over the class of objects of regularity s. In fact, a lower bound for this rate is $n^{-4s/(1+4s)}$ for $0 < s \leq 1/4$. As for the upper bound, using a Taylor expansion, it can be seen that it is enough to provide an estimate for the case $\Phi(x) = x^3$. That is the aim of this paper. Our method makes intensive use of special algebraic and wavelet properties of the Haar basis.
Publié le : 1996-04-14
Classification:
Minimax estimation,
estimation of nonlinear functionals,
integral functionals of a density,
wavelet estimate,
$U$-statistic,
G2G05,
G2G20
@article{1032894450,
author = {Kerkyacharian, G\'erard and Picard, Dominique},
title = {Estimating nonquadratic functionals of a density using Haar wavelets},
journal = {Ann. Statist.},
volume = {24},
number = {6},
year = {1996},
pages = { 485-507},
language = {en},
url = {http://dml.mathdoc.fr/item/1032894450}
}
Kerkyacharian, Gérard; Picard, Dominique. Estimating nonquadratic functionals of a density using Haar wavelets. Ann. Statist., Tome 24 (1996) no. 6, pp. 485-507. http://gdmltest.u-ga.fr/item/1032894450/