Evaluation of a blind method for the estimation of Hurst´s exponent in time series

Nowadays a lot of methods for the estimation of Hurst´s coefficient (H) in time series are available. Most of them, even if very effective, need some a priori information to be applied (in particular about the stationarity of the series). We analyzed eight up-to-date methods (working both in time and in frequency domain) at work with four kinds of synthetic time series (fBm, fGn, 1/f, FARIMA) in the range 0.1≤H≤0.9. We built graphs for each method evaluating the quality of the estimation, in terms of accuracy (bias) and precision (STD) of the deviation from the expected estimation value. Beginning from that, we formulated a procedure useful for a reliable estimation of H, using these existing methods, without any assumption on the stationarity of the time series. This procedure suggests to estimate, at first, the coefficient “alpha”, spectral slope in a bi-logarithmic scale-estimator chart, next to the zero-frequency axis, of the unknown time series. Once estimated alpha, i.e. an indirect estimation of the stationarity of the series, the procedure recommends the best method for the estimation of H, depending on the stationarity value.