Zipf-Mandelbrot law

In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. Also known as the Pareto-Zipf law, it is a power-law distribution on ranked data, named after the Harvard linguistics professor George Kingsley Zipf (1902-1950) who suggested a simpler distribution called Zipf's law, and the mathematician Benoît Mandelbrot (born November 20, 1924), who subsequently generalized it.

The probability mass function is given by:


 * $$f(k;N,q,s)=\frac{1/(k+q)^s}{H_{N,q,s}}$$

where $$H_{N,q,s}$$ is given by:


 * $$H_{N,q,s}=\sum_{i=1}^N \frac{1}{(i+q)^s}$$

which may be thought of as a generalization of a harmonic number. In the limit as $$N$$ approaches infinity, this becomes the Hurwitz zeta function $$\zeta(q,s)$$. For finite $$N$$ and $$q=0$$ the Zipf-Mandelbrot law becomes Zipf's law. For infinite $$N$$ and $$q=0$$ it becomes a Zeta distribution.

Applications
The distribution of words ranked by their frequency in a random corpus of writing is generally a power-law distribution, known as Zipf's law.

If one plots the frequency rank of words contained in a large corpus of text data versus the number of occurrences or actual frequencies, one obtains a power-law distribution, with exponent close to one (but see Gelbukh and Sidorov 2001).

References and links

 * Reprinted as
 * Z. K. Silagadze: Citations and the Zipf-Mandelbrot's law
 * NIST: Zipf's law
 * W. Li's References on Zipf's law
 * Gelbukh and Sidorov 2001: Zipf and Heaps Laws’ Coefficients Depend on Language
 * Gelbukh and Sidorov 2001: Zipf and Heaps Laws’ Coefficients Depend on Language

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