Yule-Simon Distribution

A Yule-Simon random variable with parameter $\alpha>0$ can be represented as a mixture of exponential random variates. To see this write $W$ as an exponential random variate with rate $\rho$ and a Geometric random variate $K$ with probability $1-exp(-W)$ then $K$ marginally has a Yule-Simon distribution. The latent variable representation described above is used for random variate generation.

$$\begin{eqnarray*} p \left( k; \alpha \right) & = & \alpha \frac{\Gamma\left(k\right)\Gamma\left(\alpha + 1\right)}{\Gamma\left(k+\alpha+1\right)} \\ F \left( k; \alpha \right) & = & 1 - \frac{ k \Gamma\left(k\right)\Gamma\left(\alpha + 1\right)}{\Gamma\left(k+\alpha+1\right)} \end{eqnarray*}$$

for $k = 1,2,...$.

Now

$$\begin{eqnarray*} \mu & = & \frac{\alpha}{\alpha-1}\\ \mu_{2} & = & \frac{\alpha^2}{\left(\alpha-1\right)^2\left( \alpha - 2 \right)}\\ \gamma_{1} & = & \frac{ \sqrt{\left( \alpha - 2 \right)} \left( \alpha + 1 \right)^2}{ \alpha \left( \alpha - 3 \right)}\\ \gamma_{2} & = & \frac{ \left(\alpha + 3\right) + \left(\alpha^3 - 49\alpha - 22\right)}{\alpha \left(\alpha - 4\right)\left(\alpha - 3 \right) } \end{eqnarray*}$$

for $\alpha>1$ otherwise the mean is infinite and the variance does not exist. For the variance, $\alpha>2$ otherwise the variance does not exist. Similarly, for the skewness and kurtosis to be finite, $\alpha>3$ and $\alpha>4$ respectively.

Implementation: scipy.stats.yulesimon