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

TL;DR

This paper demonstrates that maximum likelihood estimators can reliably fit power-law exponents for data in bounded sample spaces, overcoming previous limitations for exponents less than minus one.

Contribution

It derives a universal ML estimator for power-law exponents applicable to bounded discrete and continuous data, with implementation guidance and code.

Findings

ML estimators work for all exponents in bounded spaces

The estimator performs well compared to previous methods

Provides practical tools and code for data analysis

Abstract

It has been repeatedly stated that maximum likelihood (ML) estimates of exponents of power-law distributions can only be reliably obtained for exponents smaller than minus one. The main argument that power laws are otherwise not normalizable, depends on the underlying sample space the data is drawn from, and is true only for sample spaces that are unbounded from above. Here we show that power-laws obtained from bounded sample spaces (as is the case for practically all data related problems) are always free of such limitations and maximum likelihood estimates can be obtained for arbitrary powers without restrictions. Here we first derive the appropriate ML estimator for arbitrary exponents of power-law distributions on bounded discrete sample spaces. We then show that an almost identical estimator also works perfectly for continuous data. We implemented this ML estimator and discuss its performance with previous attempts. We present a general recipe of how to use these estimators and present the associated computer codes.