Yule–Simon distribution

Probability distribution
name =Yule–Simon
type =mass
pdf_

Yule–Simon PMF on a log-log scale. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
cdf_

Yule–Simon CMF. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
parameters =$ho>0,$ shape (real)
support =$k\; in\; \{1,2,dots\},$
pdf =$ho,mathrm\{B\}(k,\; ho+1),$
cdf =$1\; -\; k,mathrm\{B\}(k,\; ho+1),$
mean =$frac\{\; ho\}\{\; ho-1\},$ for $ho>1,$
median =
mode =$1,$
variance =$frac\{\; ho^2\}\{(\; ho-1)^2;(\; ho-2)\},$ for $ho>2,$
skewness =$frac\{(\; ho+1)^2;sqrt\{\; ho-2${( ho-3); ho}, for $ho>3,$
kurtosis =$ho+3+frac\{11\; ho^3-49\; ho-22\}\; \{(\; ho-4);(\; ho-3);\; ho\},$ for $ho>4,$
entropy =
mgf =$frac\{\; ho\}\{\; ho+1\};\{\}\_2F\_1(1,1;\; ho+2;\; e^t),e^t\; ,$
char =$frac\{\; ho\}\{\; ho+1\};\{\}\_2F\_1(1,1;\; ho+2;\; e^\{i,t\}),e^\{i,t\}\; ,$
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"cite journal
last = Simon
first = H. A.
title = On a class of skew distribution functions
journal = Biometrika
volume = 42
pages = 425–440
date = 1955] .

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

:$f(k;\; ho)\; =\; ho,mathrm\{B\}(k,\; ho+1),\; ,$

for integer $k\; geq\; 1$ and real $ho\; >\; 0$, where $mathrm\{B\}$ is the beta function. Equivalently the pmf can be written in terms of the falling factorial as

:$f(k;\; ho)\; =\; frac\{\; ho,Gamma(\; ho+1)\}\{(k+\; ho)^\{underline\{\; ho+1\}\; ,,$

where $Gamma$ is the gamma function. Thus, if $ho$ is an integer,

:$f(k;\; ho)\; =\; frac\{\; ho,\; ho!,(k-1)!\}\{(k+\; ho)!\}\; .,$

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

:$f(k;\; ho)\; approx\; frac\{\; ho,Gamma(\; ho+1)\}\{k^\{\; ho+1\; propto\; frac\{1\}\{k^\{\; ho+1\; .,$

This means that the tail of the Yule–Simon distribution is a realization of Zipf's law: $f(k;\; ho)$ 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 arose originally as the limiting distribution of a particular stochastic process studied by Yule as a model for the distribution of biological taxa and subtaxacite journal
last = Yule
first = G. U.
title = A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S.
journal = Philosophical Transactions of the Royal Society of London, Ser. B
volume = 213
pages = 21–87
date = 1925] . Simon dubbed this process the "Yule process" but it is more commonly known today as a preferential attachment process. The preferential attachment process is an urn process in which balls are added to a growing number of urns, each ball being allocated to an urn with probability linear in the number the urn already contains.

The distribution also arises as a continuous mixture of geometric distributions. Specifically, assume that $W$ follows an exponential distribution with scale $1/\; ho$ or rate $ho$:

:$W\; sim\; mathrm\{Exponential\}(\; ho),$:$h(w;\; ho)\; =\; ho\; ,\; exp(-\; ho,w),$

Then a Yule–Simon distributed variable $K$ has the following geometric distribution:

:$K\; sim\; mathrm\{Geometric\}(exp(-W)),$

The pmf of a geometric distribution is

:$g(k;\; p)\; =\; p\; ,\; (1-p)^\{k-1\},$

for $kin\{1,2,dots\}$. The Yule–Simon pmf is then the following exponential-geometric mixture distribution:

:$f(k;\; ho)\; =\; int\_0^\{infty\}\; ,,,\; g(k;exp(-w)),h(w;\; ho),dw,$

Generalizations

The two-parameter generalization of the original Yule distribution replaces the beta function with an incomplete beta function. The probability mass function of the generalized Yule–Simon("ρ", "α") distribution is defined as

:$f(k;\; ho,alpha)\; =\; frac\{\; ho\}\{1-alpha^\{\; ho\; ;\; mathrm\{B\}\_\{1-alpha\}(k,\; ho+1)\; ,\; ,$

with . For $alpha\; =\; 0$ the ordinary Yule–Simon("ρ") distribution is obtained as a special case. The use of the incomplete beta function has the effect of introducing an exponential cutoff in the upper tail.

ee also

* Beta function
* Preferential attachment

Bibliography

* 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".")

References