The paper studies the behaviour of three different R/S Statistic algorithms for Hurst-index estimation in self-similar and long-range dependent discrete time series. The accuracy of the algorithms under convergence and aggregation in time is obtained and compared. The results show that static blocks implementations of the R/S Statistic present better accuracy than those based on a dynamic blocks implementation. The accuracy is obtained with the use of synthetic fGn and FARIMA(0, d, 0) self-similar traces. The behavior of the algorithms for well-known LAN traces is also accomplished. Identification of current software tools and fields of science using the dynamic blocks approach is also accomplished.

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Statistical analysis and synthesis of self-similar discrete time signals are presented. The analysis equation is formally defined through a special family of basis functions of which the simplest case matches the Haar wavelet. The original discrete time series is synthesized without loss by a linear combination of the basis functions after some scaling, displacement, and phase shift. The decomposition is then used to synthesize a new second-order self-similar signal with a different Hurst index than the original. The components are also used to describe the behavior of the estimated mean and variance of self-similar discrete time series. It is shown that the sample mean, although it is unbiased, provides less information about the process mean as its Hurst index is higher. It is also demonstrated that the classical variance estimator is biased and that the widely accepted aggregated variance-based estimator of the Hurst index results biased not due to its nature (which is being unbiased and has minimal variance) but to flaws in its implementation. Using the proposed decomposition, the correct estimation of the Variance Plot is described, as well as its close association with the popular Logscale Diagram.