Fitting power-laws in empirical data with estimators that work for all exponents

Area of Science:

• Statistical Modeling
• Data Analysis
• Complex Systems

Background:

• Standard maximum likelihood (ML) methods for power-law exponent estimation are often limited to exponents less than -1.
• The normalizability of power laws depends on the sample space; unbounded spaces pose limitations not present in bounded ones.

Purpose of the Study:

• To derive a robust ML estimator for power-law exponents applicable to bounded discrete sample spaces.
• To adapt and validate this estimator for continuous data.
• To provide a practical guide and computational tools for applying the new estimator.

Main Methods:

• Derivation of a novel ML estimator for power-law distributions on bounded discrete sample spaces.
• Demonstration of the estimator's efficacy with continuous data.
• Implementation and performance analysis of the derived ML estimator.

Main Results:

• A new ML estimator is presented, capable of estimating arbitrary power-law exponents in bounded discrete sample spaces.
• The derived estimator performs effectively for continuous data as well.
• The study provides a general methodology and associated code for utilizing these estimators.

Conclusions:

• The developed ML estimator overcomes previous restrictions, enabling accurate power-law exponent estimation for bounded data.
• This method offers a reliable tool for analyzing power-law distributions in various real-world datasets.
• The findings facilitate more accurate modeling and understanding of phenomena exhibiting power-law behavior.