Hostname: page-component-7dd5485656-bvgqh Total loading time: 0 Render date: 2025-10-27T15:05:25.245Z Has data issue: false hasContentIssue false
  • English
  • Français

Taylor's law, via ratios, for some distributions with infinite mean

Part of: Limit theorems Distribution theory Applications Distribution theory - Probability

Published online by Cambridge University Press:  15 September 2017

Mark Brown ,
Joel E. Cohen  and
Victor H. de la Peña
Mark Brown*
Affiliation:
Columbia University
Joel E. Cohen*
Affiliation:
The Rockefeller University and Columbia University
Victor H. de la Peña*
Affiliation:
Columbia University
*
* Postal address: Department of Statistics, Columbia University, New York, NY 10027, USA.
** Postal address: Laboratory of Populations, The Rockefeller University, New York, NY 10065, USA. Email address: cohen@mail.rockefeller.edu
* Postal address: Department of Statistics, Columbia University, New York, NY 10027, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Taylor's law (TL) originated as an empirical pattern in ecology. In many sets of samples of population density, the variance of each sample was approximately proportional to a power of the mean of that sample. In a family of nonnegative random variables, TL asserts that the population variance is proportional to a power of the population mean. TL, sometimes called fluctuation scaling, holds widely in physics, ecology, finance, demography, epidemiology, and other sciences, and characterizes many classical probability distributions and stochastic processes such as branching processes and birth-and-death processes. We demonstrate analytically for the first time that a version of TL holds for a class of distributions with infinite mean. These distributions, a subset of stable laws, and the associated TL differ qualitatively from those of light-tailed distributions. Our results employ and contribute to the methodology of Albrecher and Teugels (2006) and Albrecher et al. (2010). This work opens a new domain of investigation for generalizations of TL.

Keywords

Taylor's lawstable distributionLaplace transformTweedie distributionheavy tailcumulant generating functionapproximate exponentialitylog linear regression

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 62P12: Applications to environmental and related topics 62P35: Applications to physics

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 657 - 669
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Albrecher, H. and Teugels, J. L. (2006). Asymptotic analysis of a measure of variation. Theory Prob. Math. Statist. 74, 1–9. CrossRefGoogle Scholar
Albrecher, H., Ladoucette, S. A. and Teugels, J. L. (2010). Asymptotics of the sample coefficient of variation and the sample dispersion. J. Statist. Planning Inf. 140, 358–368. CrossRefGoogle Scholar
Arnold, S. F. (1990). Mathematical Statistics. Prentice-Hall, Englewood Cliffs, NJ. Google Scholar
Barndorff-Nielsen, O. (2014). Information and Exponential Families in Statistical Theory. John Wiley, Chichester. CrossRefGoogle Scholar
Billingsley, P. (1986). Probability and Measure, 2nd edn. John Wiley, New York. Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press. CrossRefGoogle Scholar
Çinlar, E. (2011). Probability and Stochastics. Springer, New York. CrossRefGoogle Scholar
Cohen, J. E. and Xu, M. (2015). Random sampling of skewed distributions implies Taylor's power law of fluctuation scaling. Proc. Nat. Acad. Sci. USA 112, 7749–7754. CrossRefGoogle ScholarPubMed
de la Peña, V. H. and Yang, M. (1999). Probabilistic criteria for the order of the moment of the first passage time. Preprint, Department of Statistics, Columbia University. Google Scholar
Eisler, Z., Bartos, I. and Kertész, J. (2008). Fluctuation scaling in complex systems: Taylor's law and beyond. Adv. Physics 57, 89–142. CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York. Google Scholar
Fuchs, A. and Joffe, A. (1997). Exact formula for the expectation of the ratio of the sum of squares by the square of the sum. C. R. Acad. Sci. Paris Ser. I Math. 325, 907–909. CrossRefGoogle Scholar
Giné, E., Götze, F. and Mason, D. M. (1997). When is the Student t-statistic asymptotically standard normal? Ann. Prob. 25, 1514–1531. CrossRefGoogle Scholar
Jørgensen, B. (1987). Exponential dispersion models. J. R. Statist. Soc. B 49, 127–162. Google Scholar
Jørgensen, B. (1997). The Theory of Dispersion Models. Chapman & Hall, London. Google Scholar
Jørgensen, B., Martínez, J. R. and Vinogradov, V. (2009). Domains of attraction to Tweedie distributions. Lithuanian Math. J. 49, 399–425. CrossRefGoogle Scholar
Karamata, J. (1931). Neuer Beweis und Verallgemeinerung der Tauberschen Sätze, welche die Laplacesche und Stieltjessche Transformation betreffen. J. Reine Angew. Math. 164, 27–39. CrossRefGoogle Scholar
Kendal, W. S. (2004). Taylor's ecological power law as a consequence of scale invariant exponential dispersion models. Ecological Complexity 1, 193–209. CrossRefGoogle Scholar
Kendal, W. S. (2013). Fluctuation scaling and 1/f noise: shared origins from the Tweedie family of statistical distributions. J. Basic Appl. Physics 2, 40–49. CrossRefGoogle Scholar
Kendal, W. S. and Jørgensen, B. (2011a). Taylor's power law and fluctuation scaling explained by a central-limit-like convergence. Phys. Rev. E 83, 066115. CrossRefGoogle ScholarPubMed
Kendal, W. S. and Jørgensen, B. (2011b). Tweedie convergence: a mathematical basis for Taylor's power law, 1/f noise, and multifractality. Phys. Rev. E 84, 066120. CrossRefGoogle ScholarPubMed
Neter, J., Wasserman, W. and Kutner, M. H. (1990). Applied Linear Statistical Models: Regression, Analysis of Variance, and Experimental Designs, 3rd edn. Irwin, Homewood, IL. Google Scholar
Ross, S. M. (2002). Probability Models for Computer Science. Harcourt Academic Press, San Diego, CA. Google Scholar
Taylor, L. R. (1961). Aggregation, variance and the mean. Nature 189, 732–735. CrossRefGoogle Scholar
Tweedie, M. C. K. (1946). The regression of the sample variance on the sample mean. J. London Math. Soc. 21, 22–28. CrossRefGoogle Scholar
Tweedie, M. C. K. (1947). Functions of a statistical variate with given means, with special reference to Laplacian distributions. Proc. Camb. Phil. Soc. 43, 41–49. CrossRefGoogle Scholar