Power law distribution and multi-scale analysis in Earth sciences, finance, and other fields: Some guidelines to parameter estimation.

Power-law distributions, with their interdisciplinary applications in fractals, non-linear systems, chaos theory, self-organized criticality, and scale-free systems, have garnered significant attention in recent decades. These theories find applications across various scientific disciplines, from physics to Earth sciences, social sciences, economics, and finance. Parameter estimation for such distributions can be effectively conducted by examining data at multiple scales of observation. This article illustrates practical multi-scale analysis methods through case studies from the statistical analysis of rock fractures and financial data, explaining their advantages and the underlying hypotheses and theories. Furthermore, a novel version of a maximum likelihood-based parameter estimation criterion, adapted for multi-scale samples, is presented, reassuring the audience about its applicability.