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Fisk (Log Logistic) Distribution¶
Special case of the Burr distribution with \(d=1\). There is are one shape parameter \(c > 0\) and the support is \(x \in [0,\infty)\).
\begin{eqnarray*}\textrm{Let }k & = & \Gamma\left(1-\frac{2}{c}\right)\Gamma\left(\frac{2}{c}+1\right)-\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+1\right)\\
f\left(x;c,d\right) & = & \frac{cx^{c-1}}{\left(1+x^{c}\right)^{2}} \\
F\left(x;c,d\right) & = & \left(1+x^{-c}\right)^{-1}\\
G\left(q;c,d\right) & = & \left(q^{-1}-1\right)^{-1/c}\\
\mu & = & \Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+1\right)\\
\mu_{2} & = & k\\
\gamma_{1} & = & \frac{1}{\sqrt{k^{3}}}\left[2\Gamma^{3}\left(1-\frac{1}{c}\right)\Gamma^{3}\left(\frac{1}{c}+1\right)+\Gamma\left(1-\frac{3}{c}\right)\Gamma\left(\frac{3}{c}+1\right)\right.\\ & & \left.-3\Gamma\left(1-\frac{2}{c}\right)\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+1\right)\Gamma\left(\frac{2}{c}+1\right)\right]\\
\gamma_{2} & = & -3+\frac{1}{k^{2}}\left[6\Gamma\left(1-\frac{2}{c}\right)\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+1\right)\Gamma\left(\frac{2}{c}+1\right)\right.\\ & & -3\Gamma^{4}\left(1-\frac{1}{c}\right)\Gamma^{4}\left(\frac{1}{c}+1\right)+\Gamma\left(1-\frac{4}{c}\right)\Gamma\left(\frac{4}{c}+1\right)\\ & & \left.-4\Gamma\left(1-\frac{3}{c}\right)\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+1\right)\Gamma\left(\frac{3}{c}+1\right)\right]\\
m_{d} & = & \left(\frac{c-1}{c+1}\right)^{1/c}\, \text{if }c>1, \text{otherwise } 0\\
m_{n} & = & 1\\
h\left[X\right] & = & 2-\log c\end{eqnarray*}
Implementation: scipy.stats.fisk