Estimating nonquadratic functionals of a density using Haar wavelets. (English) Zbl 0860.62033

Kerkyacharian, Gérard; Picard, Dominique

Ann. Stat. 24, No. 2, 485-507 (1996).

Summary: 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.

MSC:

62G07	Density estimation
62G20	Asymptotic properties of nonparametric inference
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

Besov spaces; minimax estimation; estimation of nonlinear functionals; integral functionals of densities; wavelet estimate; U-statistic; rate of convergence; Taylor expansion; Haar basis

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