Wavelet based estimator for fractional Brownian motion: An experimental point of view

Wavelet based estimators of the H parameter for fractional Brownian motion (fBm) is known to have interesting asymptotical properties. In this communication, we are studying the practical point of view by testing this estimator on finite length fBm signals of N samples. These signals are generated using the circulant embedding method (CEM). CEM is fast and exact such that the synthesis of true fBm signals of long size is easy. Results show that above N=4096 wavelet based estimator is unbiased and very close to the Cramer-Rao lower bound. At the light of this study, and by combining it to recent results, a practical user guide to estimate the H parameter of fBm can be provided: if N is lower than 512 classical maximum likelihood (ML) should be chosen. For N in the interval [512,4096] Whittle ML is to be preferred. For N above 4096 wavelet based should be selected. Finally, a precise confidence interval of the true H parameter can be given.