Fast and unbiased estimator of the time-dependent Hurst exponent

We combine two existing estimators of the local Hurst exponent to improve both the goodness of fit and the computational speed of the algorithm. An application with simulated time series is implemented, and a Monte Carlo simulation is performed to provide evidence of the improvement. The estimation of the Hurst exponent of a time series is a recurring problem of great interest in many fields: finance, biology, hydrology, ecology, and signal processing, to quote a few. Since most estimators are asymptotic, large data samples are needed to obtain reliable estimates, and when the exponent changes through time, the estimates fail to capture the dynamics timely. In order to overcome this limit, we combine two techniques built using the quadratic variation estimators: the former is unbiased but displays a large variance; the latter, with a low variance, exhibits a bias which can be corrected at the cost of a computationally intensive procedure that slows down the estimation. Our simple and effective idea is to shift the biased estimates of the average difference between the unbiased and biased sequences. This removes the bias and improves the speed of the algorithm since it reduces the computations that must be carried out. Monte Carlo simulations for different sequences of the Hurst functional parameter show that (a) the estimates improve in almost all the cases considered and (b) our approach is computationally very efficient (the estimation time reduces drastically).