Parameter estimation for power-law distributions by maximum likelihood methods

Distributions following a power-law are an ubiquitous phenomenon. Methods for determining the exponent of a power-law tail by graphical means are often used in practice but are intrinsically unreliable. Maximum likelihood estimators for the exponent are a mathematically sound alternative to graphical methods.

💡 Research Summary

The paper addresses the pervasive problem of estimating the exponent (α) of power‑law distributions, which appear in fields ranging from seismology to network science. Traditional practice relies on visual inspection of log‑log plots and linear regression to obtain α, but this approach suffers from several systematic issues: (1) the discrete nature of many data sets introduces bias; (2) statistical fluctuations due to finite sample size lead to unstable estimates; (3) the choice of the lower cutoff x_min is often arbitrary, causing the tail exponent to be either under‑ or over‑estimated.

To overcome these shortcomings, the authors propose a maximum‑likelihood estimation (MLE) framework. Assuming a continuous power‑law probability density f(x) = (α − 1) x_min^{α‑1} x^{−α} for x ≥ x_min, the log‑likelihood for n observations {x_i} is

L(α) = Σ_{i=1}^{n}