Editing Power law (section)

===Estimating the exponent from empirical data===
There are many ways of estimating the value of the scaling exponent for a power-law tail, however not all of them yield [[Maximum likelihood estimation#Second-order efficiency after correction for bias|unbiased and consistent answers]]. Some of the most reliable techniques are often based on the method of [[maximum likelihood estimation|maximum likelihood]]. Alternative methods are often based on making a linear regression on either the log–log probability, the log–log cumulative distribution function, or on log-binned data, but these approaches should be avoided as they can all lead to highly biased estimates of the scaling exponent.{{sfn|Clauset|Shalizi|Newman|2009}}

====Maximum likelihood====

For real-valued, [[independent and identically distributed]] data, we fit a power-law distribution of the form

: <math>p(x) = \frac{\alpha-1}{x_\min} \left(\frac{x}{x_\min}\right)^{-\alpha}</math>

to the data <math>x\geq x_\min</math>, where the coefficient <math>\frac{\alpha-1}{x_\min}</math> is included to ensure that the distribution is [[Normalizing constant|normalized]].  Given a choice for <math>x_\min</math>, the log likelihood function becomes:

:<math>\mathcal{L}(\alpha)=\log  \prod _{i=1}^n \frac{\alpha-1}{x_\min} \left(\frac{x_i}{x_\min}\right)^{-\alpha}</math> 
The maximum of this likelihood is found by differentiating with respect to parameter <math>\alpha</math>, setting the result equal to zero. Upon rearrangement, this yields the estimator equation:

:<math>\hat{\alpha} = 1 + n \left[ \sum_{i=1}^n \ln \frac{x_i}{x_\min} \right]^{-1}</math>

where <math>\{x_i\}</math> are the <math>n</math> data points <math>x_{i}\geq x_\min</math>.<ref name=Newman/><ref name=Hall/> This estimator exhibits a small finite sample-size bias of order <math>O(n^{-1})</math>, which is small when ''n''&nbsp;>&nbsp;100. Further, the standard error of the estimate is <math>\sigma = \frac{\hat{\alpha}-1}{\sqrt{n}} + O(n^{-1})</math>. This estimator is equivalent to the popular{{citation needed|date=June 2012}} [[Hill estimator]] from [[quantitative finance]] and [[extreme value theory]].{{citation needed|date=June 2012}}

For a set of ''n'' integer-valued data points <math>\{x_i\}</math>, again where each <math>x_i\geq x_\min</math>, the maximum likelihood exponent is the solution to the transcendental equation

: <math>\frac{\zeta'(\hat\alpha,x_\min)}{\zeta(\hat{\alpha},x_\min)} = -\frac{1}{n} \sum_{i=1}^n \ln \frac{x_i}{x_\min} </math>

where <math>\zeta(\alpha,x_{\mathrm{min}})</math> is the [[Riemann zeta function#Generalizations|incomplete zeta function]]. The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for <math>\hat{\alpha}</math> are not equivalent, and the continuous version should not be applied to discrete data, nor vice versa.

Further, both of these estimators require the choice of <math>x_\min</math>. For functions with a non-trivial <math>L(x)</math> function, choosing <math>x_\min</math> too small produces a significant bias in <math>\hat\alpha</math>, while choosing it too large increases the uncertainty in <math>\hat{\alpha}</math>, and reduces the [[statistical power]] of our model. In general, the best choice of <math>x_\min</math> depends strongly on the particular form of the lower tail, represented by <math>L(x)</math> above.

More about these methods, and the conditions under which they can be used, can be found in .{{sfn|Clauset|Shalizi|Newman|2009}} Further, this comprehensive review article provides [http://www.santafe.edu/~aaronc/powerlaws/ usable code] (Matlab, Python, R and C++) for estimation and testing routines for power-law distributions.

====Kolmogorov–Smirnov estimation====

Another method for the estimation of the power-law exponent, which does not assume [[independent and identically distributed]] (iid) data, uses the minimization of the [[Kolmogorov–Smirnov statistic]], <math>D</math>, between the cumulative distribution functions of the data and the power law:

: <math>\hat{\alpha} = \underset{\alpha}{\operatorname{arg\,min}} \, D_\alpha </math>

with

: <math> D_\alpha = \max_x | P_\mathrm{emp}(x) - P_\alpha(x) | </math>

where <math>P_\mathrm{emp}(x)</math> and <math>P_\alpha(x)</math> denote the cdfs of the data and the power law with exponent <math>\alpha</math>, respectively. As this method does not assume iid data, it provides an alternative way to determine the power-law exponent for data sets in which the temporal correlation can not be ignored.<ref name=Klaus/>

====Two-point fitting method====
This criterion<ref>{{Cite journal |last1=Guerriero |first1=Vincenzo |last2=Vitale |first2=Stefano |last3=Ciarcia |first3=Sabatino |last4=Mazzoli |first4=Stefano |date=2011-05-09 |title=Improved statistical multi-scale analysis of fractured reservoir analogues |url=https://www.sciencedirect.com/science/article/pii/S0040195111000047 |journal=Tectonophysics |language=en |volume=504 |issue=1 |pages=14–24 |doi=10.1016/j.tecto.2011.01.003 |bibcode=2011Tectp.504...14G |issn=0040-1951}}</ref> can be applied for the estimation of power-law exponent in the case of scale-free distributions and provides a more convergent estimate than the maximum likelihood method. It has been applied to study probability distributions of fracture apertures. In some contexts the probability distribution is described, not by the [[cumulative distribution function]], by the [[cumulative frequency analysis|cumulative frequency]] of a property ''X'', defined as the number of elements per meter (or area unit, second etc.) for which ''X''&nbsp;>&nbsp;''x'' applies, where ''x'' is a variable real number. As an example,{{Cn|date=November 2019}} the cumulative distribution of the fracture aperture, ''X'', for a sample of ''N'' elements is defined as 'the number of fractures per meter having aperture greater than ''x'' . Use of cumulative frequency has some advantages, e.g. it allows one to put on the same diagram data gathered from sample lines of different lengths at different scales (e.g. from outcrop and from microscope).