Application of resampling and linear spline methods to spectral and dispersional analyses of long-memory processes

Most methods for estimating the Hurst exponent H   of long-memory processes (R/SR/S analysis, correlogram plot, dispersional analysis, periodogram plot) are based on linear regression. All of these methods share a common drawback: the regression line should be fitted on an unknown part of the data. A general remedy to this problem is proposed, consisting in fitting a linear spline with an unknown breakpoint, instead of the usual regression line. From another side, in the case of partition-based methods such as dispersional analysis, the number of points available for regression is generally small. It is shown that this number can be increased by drawing well-suited random sub-series. The proposed improvements are first tested on simulated fractional Gaussian noises. Then, two datasets are processed: the Nile series and the North Atlantic Oscillation annual index series. The former is long enough (663 values) for investigating its long memory, and we found estimates of H   close to those already published. The latter is much shorter (141 values). As a whole, it seems persistent, with a Hurst coefficient stronger than estimations (about 0.640.64) found in the literature. However, it is shown that this phenomenon could be non-stationary, with H   switching from 0.50.5 to 0.80.8 in the thirties.