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 parameters of the distribution.