  We prove the consistency of the Power-Law Fit PLFit method proposed by
Clauset et al.(2009) to estimate the power-law exponent in data coming from a
distribution function with regularly-varying tail. In the complex systems
community, PLFit has emerged as the method of choice to estimate the power-law
exponent. Yet, its mathematical properties are still poorly understood.
  The difficulty in PLFit is that it is a minimum-distance estimator. It first
chooses a threshold that minimizes the Kolmogorov-Smirnov distance between the
data points larger than the threshold and the Pareto tail, and then applies the
Hill estimator to this restricted data. Since the number of order statistics
used is random, the general theory of consistency of power-law exponents from
extreme value theory does not apply. Our proof consists in first showing that
the Hill estimator is consistent for general intermediate sequences for the
number of order statistics used, even when that number is random. Here, we call
a sequence intermediate when it grows to infinity, while remaining much smaller
than the sample size. The second, and most involved, step is to prove that the
optimizer in PLFit is with high probability an intermediate sequence, unless
the distribution has a Pareto tail above a certain value. For the latter
special case, we give a separate proof.
