Parabolic fractal distribution

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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.

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.; Yang, Y. G.; Bao, Z. Y.; Ke, X. Z.; Liu, X. L. (2009). "Mineral resource analysis by parabolic fractals". Mining Science and Technology (China) 19: 91–19. doi:10.1016/S1674-5264(09)60017-X. edit

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf-Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding-Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse multivariate gamma inverse-Wishart matrix normal matrix t multivariate gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Parabolic fractal distribution

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