International Journal of Wavelets, Multiresolution and Information Processing c © World Scientific Publishing Company NEW INSIGHTS INTO THE ESTIMATION OF SCALING EXPONENTS ∗

We study the statistical performance of multiresolution-based estimation procedures for the scaling exponents of multifractal processes. These estimators rely on the computa-tion of multiresolution quantities such as wavelet, increment or aggregation coefficients. Estimates are obtained by linear fits performed in log of structure functions of order q versus log of scale plots. Using various and recent types of multiplicative cascades and a large variety of multifractal processes, we study and benchmark, by means of numerical simulations, the statistical performance of these estimation procedures. We show that they all undergo a systematic linearisation effect: for a range of orders q, the estimates account correctly for the scaling exponents; outside that range, the estimates signifi-cantly depart from the correct values and systematically behave as linear functions of q. The definition and characterisation of this effect are thoroughly studied. In contra-diction with interpretations proposed in the literature, we provide numerical evidence leading to the conclusion that this linearisation effect is neither a finite size effect nor a infiniteness of moments effect, but that its origin should be related to the deep nature of the process itself. We comment on its importance and consequences for the practical analysis of the multifractal properties of empirical data