Maximum Likelihood Estimation of Power-Law Exponents for Testing Universality in Complex Systems

Power-law-type distributions are extensively found when studying the behavior of many complex systems. However, due to limitations in data acquisition, empirical datasets often only cover a narrow range of observations, making it difficult to establish power-law behavior unambiguously. In this work, we present a statistical procedure to merge different datasets, with two different aims. First, we obtain a broader fitting range for the statistics of different experiments or observations of the same system. Second, we establish whether two or more different systems may belong to the same universality class. By means of maximum likelihood estimation, this methodology provides rigorous statistical information to discern whether power-law exponents characterizing different datasets can be considered equal to each other or not. This procedure is applied to the Gutenberg–Richter law for earthquakes and for synthetic earthquakes (acoustic emission events) generated in the laboratory: labquakes (Navas-Portella et al. Phys Rev E 100:062106, 2019).