Power-law distributions in empirical data. (English) Zbl 1176.62001

Clauset, Aaron; Shalizi, Cosma Rohilla; Newman, M. E. J.

SIAM Rev. 51, No. 4, 661-703 (2009).

Summary: Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distributions – the part of the distributions representing large but rare events – and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all.
We present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.

MSC:

62-07	Data analysis (statistics) (MSC2010)
62G10	Nonparametric hypothesis testing
65C60	Computational problems in statistics (MSC2010)
62P99	Applications of statistics
62F99	Parametric inference

Keywords:

power-law distributions; Pareto; Zipf; maximum likelihood; heavy-tailed distributions; likelihood ratio test; model selection

Software:

plfit