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February 2005 Strong, Weak and False Inverse Power Laws
Richard Perline
Statist. Sci. 20(1): 68-88 (February 2005). DOI: 10.1214/088342304000000215


Pareto, Zipf and numerous subsequent investigators of inverse power distributions have often represented their findings as though their data conformed to a power law form for all ranges of the variable of interest. I refer to this ideal case as a strong inverse power law (SIPL). However, many of the examples used by Pareto and Zipf, as well as others who have followed them, have been truncated data sets, and if one looks more carefully in the lower range of values that was originally excluded, the power law behavior usually breaks down at some point. This breakdown seems to fall into two broad cases, called here (1) weak and (2) false inverse power laws (WIPL and FIPL, resp.). Case 1 refers to the situation where the sample data fit a distribution that has an approximate inverse power form only in some upper range of values. Case 2 refers to the situation where a highly truncated sample from certain exponential-type (and in particular, “lognormal-like”) distributions can convincingly mimic a power law. The main objectives of this paper are (a) to show how the discovery of Pareto–Zipf-type laws is closely associated with truncated data sets; (b) to elaborate on the categories of strong, weak and false inverse power laws; and (c) to analyze FIPLs in some detail. I conclude that many, but not all, Pareto–Zipf examples are likely to be FIPL finite mixture distributions and that there are few genuine instances of SIPLs.


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Richard Perline. "Strong, Weak and False Inverse Power Laws." Statist. Sci. 20 (1) 68 - 88, February 2005.


Published: February 2005
First available in Project Euclid: 6 June 2005

zbMATH: 1100.62013
MathSciNet: MR2182988
Digital Object Identifier: 10.1214/088342304000000215

Keywords: Extreme value theory , Gumbel distributions , Lognormal distribution , mixture distributions , Pareto distribution , Pareto–Zipf laws

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.20 • No. 1 • February 2005
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