Problems with fitting to the power-law distribution

By Michel L. Goldstein, S. Morris, Gary G. Yen. 2004

693 citations recorded by the source corpus.

Abstract.This short communication uses a simple experiment to show that fitting to a power law distribution by using graphical methods based on linear fit on the log-log scale is biased and inaccurate. It shows that using maximum likelihood estimation (MLE) is far more robust. Finally, it presents a new table for performing the Kolmogorov-Smirnov test for goodness-of-fit tailored to power-law distributions in which the power-law exponent is estimated using MLE. The techniques presented here will advance the application of complex network theory by allowing reliable estimation of power-law models from data and further allowing quantitative assessment of goodness-of-fit of proposed power-law models to empirical data.

Subfields: Economics and Econometrics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/918d186c59f1172843b0fa5530f515d0ca2505c8

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Citations in the corpus

Informetrics at the beginning of the 21st century - A review (2008, background): “The Goldstein et al. (2004) paper has already been cited extensively (24, 28 and 57 citations according to WOS, Scopus and GS, respectively), even though their findings about the usage of the maximum likelihood estimator is already known for quite some time (see, for example Nicholls, 1989).”
The Network Structure of the CDS Market and its Determinants (2014, methodology): “Goldstein et al. (2004) show that the most robust procedure to do so is the use of a maximum likelihood estimator.”
Fitting Heavy Tailed Distributions: The poweRlaw Package (2014, positive): “Functions are provided for plotting, comparing distributions and estimating parameter uncertainty.”
a, b careful: The challenge of scale invariance for comparative analyses in power law models of the streamflow recession (2015, background): “These issues have been raised in several comprehensive reviews, which also demonstrate appropriate fitting and estimation techniques that avoid such biases [Goldstein et al., 2004; Clauset et al., 2009].”
Power-Law Distributions in Empirical Data (2007, methodology): “Reference [51] and more recently [19] treated the special case xmin = 1, showing that the appropriate estimator for α is given by the solution to the transcendental equation”
On the probability of occurrence of extreme space weather events (2012, methodology): “The question is: why do we always observe MLE > LS when they should be nearly equal? Could this be attributed to some statistical feature of power laws? I had expected that MLE − LS > 0 would be the norm, partly because that was emphatically the case for the data sets (of which three of four had MLE”
Power-law distributions in binned empirical data (2012, methodology): “[As an example, see Table 2 of Goldstein, Morris and Yen (2004).]”
Repeat burglary victimisation: a tale of two theories (2008, background): “Goldstein et al. (2004) show that using the first five points of the log–log plot produces a good estimate of γ, whereas problems arise when data for the remainder of the distribution are considered.”
Fitting and goodness-of-fit test of non-truncated and truncated power-law distributions (2012, methodology): “Goldstein et al. (2004), Bauke (2007), White et al. (2008), and Clauset et al. (2009) check several methods based in the fitting of the estimated probability densities or cumulative distributions in the power-law case.”
When individual behaviour matters: homogeneous and network models in epidemiology (2007, background): “This approach, however, can be flawed, especially if the data are transformed to a log–log scale (Goldstein et al. 2004).”