Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series

Abstract

Three-scaled windowed variance methods (standard, linear regression detrended, and bridge detrended) for estimating the Hurst coefficient ( H) are evaluated. The Hurst coefficient, with 0 < H < 1, characterizes self-similar decay in the time-series autocorrelation function. The scaled windowed variance methods estimate H for fractional Brownian motion (fBm) signals which are cumulative sums of fractional Gaussian noise (fGn) signals. For all three methods both the bias and standard deviation of estimates are less than 0.05 for series having N ⩾ 2 9 points. Estimates for short series ( N < 2 8) are unreliable. To have a 0.95 probability of distinguishing between two signals with true H differing by 0.1, more than 2 15 points are needed. All three methods proved more reliable (based on bias and variance of estimates) than Hurst's rescaled range analysis, periodogram analysis, and autocorrelation analysis, and as reliable as dispersional analysis. The latter methods can only be applied to fGn or differences of fBm, while the scaled windowed variance methods must be applied to fBm or cumulative sums of fGn.