Maximum Likelihood Estimate for power laws
Author jlapeyre
2 Stars
Updated Last
3 Years Ago
Started In
May 2018


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Physicists love power laws. But, they don't always use the best methods for extracting powers from empirical data.

Notebook example

Here is a notebook using MaximumLikelihoodPower.jl (this notebook is in the Notebooks folder in this distribution).

Command-line example

import MaximumLikelihoodPower
const MLE = MaximumLikelihoodPower

julia> seed = 11; α = 0.5;

# Get 10^6 samples from the Pareto distribution
julia> data = MLE.Example.makeparetodata(α, seed);

# Minimize the Kolmogorov-Smirnov statistic
# The second value returned is the minimizing alpha
julia> MLE.scanKS(data, range(.4, length=11, stop=.6))
3-element Array{Float64,1}:



    (estimate, stderr) = mle(data::AbstractVector)

Return the maximum likelihood estimate and standard error of the exponent of a power law applied to the sorted vector data.


    KSstatistic(data::AbstractVector, alpha) --> Float64

Return the Kolmogorov-Smirnov statistic comparing data to a power law with power alpha. The elements of data are assumed to be unique. Minimizing the KS statistic over alpha is another way to estimate the parameter of the sample distribution. See testKS in the test directory.

scanKS(data, powers)

Compute the Kolmogorov Smirnov statistic for several values of α in the iterator powers. Return the value of α that minimizes the KS statistic and the two neighboring values.



Return the maximum likelihood estimate and standard error of the exponent of a power law applied to the sorted vector data. Also return the Kolmogorov-Smirnov statistic. Results are returned in an instance of type MLEKS.


    scanmle(data::AbstractVector; ntrials=100, stderrcutoff=0.1, useKS=false)

Perform mle approximately ntrials times on data, increasing xmin. Stop trials if the stderr of the estimate alpha is greater than stderrcutoff. Return an object containing statistics about the scan.


    comparescan(mle::MLEKS, i, data, mlescan::MLEScan)

compare the results of MLE estimation mle to record results in mlescan and update mlescan.


Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-Law Distributions in Empirical Data. SIAM Review, 51(4), 661–703.,