Editing Power law (section)

====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.