Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T04:17:33.824Z Has data issue: false hasContentIssue false
  • English
  • Français

SQUARED COEFFICIENT OF VARIATION OF TAYLOR'S LAW FOR RANDOM ABSOLUTE DIFFERENCES

Published online by Cambridge University Press:  19 July 2017

Mark Brown  and
Joel E. Cohen
Mark Brown
Affiliation:
Department of Statistics, Columbia University, New York, NY 10027USA E-mail: [email protected]
Joel E. Cohen
Affiliation:
Laboratory of Populations, Rockefeller University, New York, NY 10065 USA; Earth Institute and Department of Statistics, Columbia University, New York, NY 10027 USA; Department of Statistics, University of Chicago, Chicago, IL 60637USA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In a family, parameterized by θ, of non-negative random variables with finite, positive second moment, Taylor's law (TL) asserts that the population variance is proportional to a power of the population mean as θ varies: σ2 (θ) = a[μ(θ)]b, a > 0. TL, sometimes called fluctuation scaling, holds widely in science, probability theory, and stochastic processes. Here we report diverse examples of TL with b = 2 (equivalent to a constant coefficient of variation) arising from a difference of random variables in normed vector spaces of dimension 1 and larger. In these examples, we compute a exactly using, in some cases, a simple, new technique. These examples may prove useful in future models that involve differences of random variables, including models of the spatial distribution and migration of human populations.

Keywords

coefficient of variationgeometric probabilitymigrationsignal-to-noise ratioTaylor's law
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 32 , Issue 4 , October 2018 , pp. 483 - 494
Copyright
Copyright © Cambridge University Press 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.)

References

1.Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing. New York: Holt.Google Scholar
2.Cohen, J.E. & Courgeau, D. (In press). Modeling distances between humans using Taylor's law and geometric probability. Mathematical Population Studies.Google Scholar
3.Courgeau, D. (1970). Les champs migratoires en France. Paris: Presses Universitaires de France.Google Scholar
4.Courgeau, D. (1973). Migrations et découpages du territoire. Population 28(3): 511–537.Google Scholar
5.Courgeau, D. & Baccaïni, B. (1989). Migrations et distances. Population 42(1): 57–82.Google Scholar
6.Eisler, Z., Bartos, I. & Kertész, J. (2008). Fluctuation scaling in complex systems: Taylors law and beyond. Advances in Physics 57(1): 89–142. doi: 10.1080/00018730801893043.Google Scholar
7.Feller, W. (1971). An introduction to probability theory and its applications, vol. II, 2nd ed. New York: John Wiley and Sons.Google Scholar
8.Keilson, J. (1979). Markov chain models, rarity and exponentiality. New York: Springer-Verlag.Google Scholar
9.Marshall, A.W. & Olkin, I. (2007). Life distributions: structure of nonparametric, semiparametric and parametric families. Springer series in statistics. New York: Springer-Verlag.Google Scholar
10.Taylor, L.R. (1961). Aggregation, variance and the mean. Nature 189: 732–735.Google Scholar