Parabolic fractal distribution

In probability and statistics, the parabolic fractal distribution is a type of discrete probability distribution in which the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship (see references below).

In a number of applications, there is a so-called King effect where the highest-ranked item has a significantly greater frequency or size than the model predicts on the basis of the other items.

Definition

The probability mass function is given, as a function of the rank n, by

$f(n;b,c) \propto n^{-b} \exp(-c(\log n)^2) ,$

where b and c are paramerers of the distribution.

See also

• Zipf's law

References

• Laherrère J, Deheuvels P (1996) "Distributions de type 'fractal parabolique' dans la nature" (French, with English summary: "Parabolic fractal" distributions in nature), (http://www.hubbertpeak.com/laherrere/fractal.htm) Comptes rendus de l'Académie des sciences, Série II a: Sciences de la Terre et des Planétes, 322, (7), 535–541
• Xie, S. (2009). "Mineral resource analysis by parabolic fractals". Mining Science and Technology (China) 19: 91–19. doi:10.1016/S1674-5264(09)60017-X.