Hurst Parameter Estimator Based on a Decomposition by Aggregated Series

In this work, a comparison of three estimators of the Hurst parameter is presented: the classical variance-based method and two new estimators based on an orthogonal decomposition (that can be achieved by aggregated series or by using Haar filters) and that use, respectively, a weighted and a non-weighted linear regression. These three estimators were applied to a set of synthetic fGN traces. The analyses showed that the variance method and the estimator that uses non-weighted linear regression underestimate the theoretical value of H, while the third estimator, decomposition-based, that uses a weighted linear regression, shows an excellent behavior. This estimator presented a bias nearer to zero and the lowest standard error when applied to fGN traces for lengths from 1024 to 1048576 samples. The presented decomposition can be extended to study the frequency information of the time series, by obtaining what we call Hurst spectrum, and to generate time series that comply the definitions of self-similar time series.