Editing Power law (section)

===Plotting power-law distributions===
In general, power-law distributions are plotted on [[log–log plot|doubly logarithmic axes]], which emphasizes the upper tail region. The most convenient way to do this is via the (complementary) [[cumulative distribution function#Complementary cumulative distribution function (tail distribution)|cumulative distribution]] (ccdf) that is, the [[survival function]], <math>P(x) = \mathrm{Pr}(X > x)</math>,

:<math>P(x) = \Pr(X > x) =  C \int_x^\infty p(X)\,\mathrm{d}X =  \frac{\alpha-1}{x_\min^{-\alpha+1}} \int_x^\infty X^{-\alpha}\,\mathrm{d}X = \left(\frac{x}{x_\min} \right)^{-(\alpha-1)}.</math>

The cdf is also a power-law function, but with a smaller scaling exponent. For data, an equivalent form of the cdf is the rank-frequency approach, in which we first sort the <math>n</math> observed values in ascending order, and plot them against the vector <math>\left[1,\frac{n-1}{n},\frac{n-2}{n},\dots,\frac{1}{n}\right]</math>.

Although it can be convenient to log-bin the data, or otherwise smooth the probability density (mass) function directly, these methods introduce an implicit bias in the representation of the data, and thus should be avoided.{{sfn|Clauset|Shalizi|Newman|2009}}<ref>{{cite journal|title=Parameter estimation for power-law distributions by maximum likelihood methods|journal= European Physical Journal B|volume=58 |issue=2|pages=167–173|author=Bauke, H. |doi=10.1140/epjb/e2007-00219-y|year=2007|arxiv=0704.1867 |bibcode=2007EPJB...58..167B|s2cid=119602829}}</ref> The survival function, on the other hand, is more robust to (but not without) such biases in the data and preserves the linear signature on doubly logarithmic axes. Though a survival function representation is favored over that of the pdf while fitting a power law to the data with the linear least square method, it is not devoid of mathematical inaccuracy. Thus, while estimating exponents of a power law distribution, maximum likelihood estimator is recommended.