Parameter estimation for power-law distributions by maximum likelihood methods

@article{Bauke2007ParameterEF,
  title={Parameter estimation for power-law distributions 
 by maximum likelihood methods},
  author={Heiko Bauke},
  journal={The European Physical Journal B},
  year={2007},
  volume={58},
  pages={167-173}
}
  • H. Bauke
  • Published 14 April 2007
  • Mathematics
  • The European Physical Journal B
Abstract.Distributions following a power-law are an ubiquitous phenomenon. Methods for determining the exponent of a power-law tail by graphical means are often used in practice but are intrinsically unreliable. Maximum likelihood estimators for the exponent are a mathematically sound alternative to graphical methods.  

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References

SHOWING 1-10 OF 18 REFERENCES
Problems with fitting to the power-law distribution
Abstract.This short communication uses a simple experiment to show that fitting to a power law distribution by using graphical methods based on linear fit on the log-log scale is biased and
How to make a Hill Plot
An abundance of high quality data sets requiring heavy tailed models necessitates reliable methods of estimating the shape parameter governing the degree of tail heaviness. The Hill estimator is a
Introduction to Probability and Mathematical Statistics
Probability Random Variables and Their Distributions Special Probability Distributions Joint Distributions Properties of Random Variables Functions of Random Variables Limiting Distributions
In all likelihood : statistical modelling and inference using likelihood
TLDR
This paper presents a meta-modelling framework for estimating the likelihood of random parameters in a discrete-time environment and describes its use in simple and complex models.
Power laws, Pareto distributions and Zipf's law
TLDR
Some of the empirical evidence for the existence of power-law forms and the theories proposed to explain them are reviewed.
The statistical sleuth : a course in methods of data analysis
1. Drawing Statistical Conclusions. 2. Inference Using t-Distributions. 3. A Closer Look at Assumptions. 4. Alternatives to the t-Tools. 5. Comparisons among Several Samples. 6. Linear Combinations
Revisiting "scale-free" networks.
  • E. Keller
  • Computer Science
    BioEssays : news and reviews in molecular, cellular and developmental biology
  • 2005
TLDR
The real surprise, if any, is that power-law distributions are easy to generate, and by a variety of mechanisms; the architecture that results is not universal, but particular; it is determined by the actual constraints on the system in question.
A preferential attachment model with Poisson growth for scale-free networks
TLDR
A scale-free network model with a tunable power-law exponent that can generate any network is proposed, motivated by an application in Bayesian inference implemented as Markov chain Monte Carlo to estimate a network.
The European Phys ical Journal B41(2)
  • 255
  • 2004
BioEssays27(10)
  • 1060
  • 2005
...
1
2
...