Wavelet-Based Estimators of Scaling Behavior

Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are studied extensively. These estimators mainly include the (bi)orthogonal wavelet estimators and the Wavelet Transform Modulus Maxima (WTMM) estimator. This study focuses both on short and long time-series. In the framework of Fractional Auto-Regressive Integrated Moving Average (FARIMA) processes, we advocate the use of approximately adapted wavelet estimators. For these "ideal" processes, the scaling behavior actually extends down to the smallest scale, i.e., the sampling period of the time series, if an adapted decomposition is used. But in practical situations, there generally exists a cut-o# scale below which the scaling behavior no longer holds. We test the robustness of the set of wavelet-based estimators with respect to that cut-o# scale as well as to the specific density of the underlying law of the process. In all situations the WTMM estimator is shown to be the best or among the best estimators in terms of the mean square error. We also compare the wavelet estimators with the Detrended Fluctuation Analysis (DFA) estimator which was recently proved to be among the best estimators which are not wavelet-based estimators. The WTMM estimator turns out to be a very competitive estimator which can be further generalized to characterize multiscaling behavior