Yule-Simon distribution

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\,] for [\rho>3\,]| kurtosis =[\rho+3+\frac \,] for [\rho>4\,]| entropy =| mgf =[\frac\;_2F_1(1,1; \rho+2; e^t)\,e^t \,]| char =[\frac\;_2F_1(1,1; \rho+2; e^)\,e^ \,]| }} In probability and statistics, the Yule-Simon distribution is a discrete probability distribution named after Udny Yule and Herbert Simon. Simon originally called it the Yule distribution.

The probability mass function of the Yule-Simon(ρ) distribution is

[f(k;\rho) = \rho\,\mathrm(k, \rho+1), \,]

for integer [k \geq 1] and real [\rho > 0], where [\mathrm] is the beta function. Equivalently the pmf can be written in terms of the falling factorial as

[ f(k;\rho) = \frac}} ,\,]

where [\Gamma] is the gamma function. Thus, if [\rho] is an integer,

[ f(k;\rho) = \frac .\,]

The probability mass function f has the property that for sufficiently large k we have

[ f(k;\rho) \approx \frac} \propto \frac} .\,]

This means that the tail of the Yule-Simon distribution is a realization of Zipf's law: [f(k;\rho)] can be used to model, for example, the relative frequency of the [k]th most frequent word in a large collection of text, which according to Zipf's law is inversely proportional to a (typically small) power of [k].

Occurrence

The Yule-Simon distribution arises as a continuous mixture of geometric distributions. Specifically, assume that [W] follows an exponential distribution with scale [1/\rho] or rate [\rho]:

[W \sim \mathrm(\rho)\,]

[h(w;\rho) = \rho \, \exp(-\rho\,w)\,]

Then a Yule-Simon distributed variable [K] has the following geometric distribution:

[K \sim \mathrm(\exp(-W))\,]

The pmf of a geometric distribution is

[g(k; p) = p \, (1-p)^\,]

for [k\in\]. The Yule-Simon pmf is then the following exponential-geometric mixture distribution:

[f(k;\rho) = \int_0^ \,\,\, g(k;\exp(-w))\,h(w;\rho)\,dw\,]

Generalizations

Simon also hinted at a two-parameter generalization of the Yule-Simon distribution, in which the beta function is replaced by an incomplete beta function. The probability mass function of the generalized Yule-Simon(ρ, α) distribution is defined as

[ f(k;\rho,\alpha) = \frac} \; \mathrm_(k, \rho+1) , \,]

with [0 \leq \alpha < 1]. For [\alpha = 0] the ordinary Yule-Simon(ρ) distribution is obtained as a special case.

References

• Herbert A. Simon, On a Class of Skew Distribution Functions, Biometrika 42(3/4): 425–440, December 1955.
• Colin Rose and Murray D. Smith, Mathematical Statistics with Mathematica. New York: Springer, 2002, ISBN 0-387-95234-9. (See page 107, where it is called the "Yule distribution".)

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