Paper detail

Fitting and goodness-of-fit test of non-truncated and truncated power-law distributions

Power-law distributions contain precious information about a large variety of processes in geoscience and elsewhere. Although there are sound theoretical grounds for these distributions, the empirical evidence in favor of power laws has been traditionally weak. Recently, Clauset et al. have proposed a systematic method to find over which range (if any) a certain distribution behaves as a power law. However, their method has been found to fail, in the sense that true (simulated) power-law tails are not recognized as such in some instances, and then the power-law hypothesis is rejected. Moreover, the method does not work well when extended to power-law distributions with an upper truncation. We explain in detail a similar but alternative procedure, valid for truncated as well as for non-truncated power-law distributions, based in maximum likelihood estimation, the Kolmogorov-Smirnov goodness-of-fit test, and Monte Carlo simulations. An overview of the main concepts as well as a recipe for their practical implementation is provided. The performance of our method is put to test on several empirical data which were previously analyzed with less systematic approaches. The databases presented here include the half-lives of the radionuclides, the seismic moment of earthquakes in the whole world and in Southern California, a proxy for the energy dissipated by tropical cyclones elsewhere, the area burned by forest fires in Italy, and the waiting times calculated over different spatial subdivisions of Southern California. We find the functioning of the method very satisfactory.