On estimating the exponent of power-law frequency distributions
ABSTRACT Power-law frequency distributions characterize a wide array of natural phenomena. In ecology, biology, and many physical and social sciences, the exponents of these power laws are estimated to draw inference about the processes underlying the phenomenon, to test theoretical models, and to scale up from local observations to global patterns. Therefore, it is essential that these exponents be estimated accurately. Unfortunately, the binning-based methods traditionally used in ecology and other disciplines perform quite poorly. Here we discuss more sophisticated methods for fitting these exponents based on cumulative distribution functions and maximum likelihood estimation. We illustrate their superior performance at estimating known exponents and provide details on how and when ecologists should use them. Our results confirm that maximum likelihood estimation outperforms other methods in both accuracy and precision. Because of the use of biased statistical methods for estimating the exponent, the conclusions of several recently published papers should be revisited.
Full-textDOI: · Available from: Brian J. Enquist, May 29, 2015
SourceAvailable from: Bruce D. MalamudRisk and Uncertainty Assessment for Natural Hazards, Edited by J Rougier, RSJ Sparks, LJ Hill, 01/2013: chapter 12: pages 398-444; Cambridge University Press., ISBN: 978-1-10700-619-5
Oecologia Australis 06/2011; 15(2):199-212. DOI:10.4257/oeco.2011.1502.02
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ABSTRACT: Reviews of the management, entrepreneurship, and marketing literature suggest that most performance-based outcomes are not distributed according to Gaussian assumptions within the normal, bell-shaped curve. Instead, Paretian (i.e., power-law) distributions are the new norm, where extreme outliers occur far more frequently and, more importantly, have a disproportionate influence on the larger system than normal statistics would lead us to believe. The unique statistical properties of power-law distributions require a scale-free theory, where a single explanation at one level applies to multiple units at the preceding level. As such, I develop a hypothesis to suggest that a founder's expectations for future growth in the nascent organizing stage can influence a venture's potential ability to scale up into an extreme outcome at later stages. I use MATLAB to construct semi-parametric bootstrap estimates for maximum likelihood fit with a power-law model on representative sample datasets from three levels of self-organized venture emergence: nascent, active start-up, and hyper-growth. I find substantial support for the scale-free hypothesis – a universal scaling exponent of ~ 1.75 – at multiple units and levels of analysis. I use the results to suggest various implications for theory, practice, pedagogy, and policy. ACKNOWLEDGEMENTAcademy of Management, Boston; 08/2012