## The Annals of Statistics

### Estimating nonquadratic functionals of a density using Haar wavelets

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 24, Number 2 (1996), 485-507.

Dates
First available in Project Euclid: 24 September 2002

https://projecteuclid.org/euclid.aos/1032894450

Digital Object Identifier
doi:10.1214/aos/1032894450

Mathematical Reviews number (MathSciNet)
MR1394973

Zentralblatt MATH identifier
0860.62033

Subjects
Primary: G2G05 G2G20

#### Citation

Kerkyacharian, Gérard; Picard, Dominique. Estimating nonquadratic functionals of a density using Haar wavelets. Ann. Statist. 24 (1996), no. 2, 485--507. doi:10.1214/aos/1032894450. https://projecteuclid.org/euclid.aos/1032894450

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