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April 1996 Estimating nonquadratic functionals of a density using Haar wavelets
Gérard Kerkyacharian, Dominique Picard
Ann. Statist. 24(2): 485-507 (April 1996). DOI: 10.1214/aos/1032894450


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.


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Gérard Kerkyacharian. Dominique Picard. "Estimating nonquadratic functionals of a density using Haar wavelets." Ann. Statist. 24 (2) 485 - 507, April 1996.


Published: April 1996
First available in Project Euclid: 24 September 2002

zbMATH: 0860.62033
MathSciNet: MR1394973
Digital Object Identifier: 10.1214/aos/1032894450

Primary: G2G05 , G2G20

Keywords: $U$-statistic , estimation of nonlinear functionals , integral functionals of a density , minimax estimation , wavelet estimate

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 2 • April 1996
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