Yule-Simon DistributionΒΆ
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