Estimation of a smoothness parameter by spline wavelets
Volume 40 / 2013
Abstract
We consider the smoothness parameter of a function $f\in L^2(\mathbb {R})$ in terms of Besov spaces $B^s_{2,\infty }(\mathbb {R})$, \[ s^*(f)=\sup\{s>0: f\in B^s_{2,\infty }(\mathbb {R})\}. \] 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 ($p\geq 1$) 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 $(p\geq 1).$
