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Estimation of a smoothness parameter by spline wavelets

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Meller M., Jarzębkowska N.

Applicationes Mathematicae

2013

tom 40

nr 3

309-326

We consider the smoothness parameter of a function f ∈ L²(ℝ) in terms of Besov spaces $B^{s}_{2,∞}(ℝ)$, $s*(f) = sup{s > 0: f ∈ B^{s}_{2,∞}(ℝ)}$. The existing results on estimation of smoothness [K. Dziedziul, M. Kucharska and B. Wolnik, J. Nonparametric Statist. 23 (2011)] employ the Haar basis and are limited to the case 0 < s*(f) < 1/2. Using p-regular (p ≥ 1) spline wavelets with exponential decay we extend them to density functions with 0 < s*(f) < p+1/2. Applying the Franklin-Strömberg wavelet p = 1, we prove that the presented estimator of s*(f) is consistent for piecewise constant functions. Furthermore, we show that the results for the Franklin-Strömberg wavelet can be generalised to any spline wavelet (p ≥ 1).

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1 Applicationes Mathematicae

1 Jarzębkowska N.

1 Meller M.

1 2013

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Estimation of a smoothness parameter by spline wavelets