On the robustness of Hurst estimators

The presence and the nature of long-range dependent (LRD) are usually characterised by the Hurst parameter. In order to meet the requirements of analysing the LRD processes, a number of practical estimation methods have been proposed in the literature. Furthermore, some efforts have been made to evaluate the accuracy and validity of the Hurst estimators for LRD processes. In practice, however, many signals measured are corrupted with various types of noises, and sometimes even the concerned signal itself has infinite variance. In such cases, which estimator has the best robustness to the LRD property of the signal and its noise involved, and how robust it is are still unresolved. The aim of this paper is to make a quantitative analysis of the robustness of twelve commonly used Hurst parameter estimators. In this paper, we considered four types of LRD signals with possible noises. They are 1) LRD process alone; 2) LRD process corrupted by 30 dB signal to noise ratio (SNR) white Gaussian noise; 3) LRD process corrupted by 30 dB SNR stable noise; 4) fractional autoregressive moving average (FARIMA) time series with stable innovations. Moreover, the standard errors of each estimator are provided.