This is documentation for an old release of SciPy (version 1.5.4). Read this page in the documentation of the latest stable release (version 1.15.1).
Wrapped Cauchy Distribution¶
There is one shape parameter \(c\in\left(0,1\right)\) with support \(x\in\left[0,2\pi\right]\).
\begin{eqnarray*} f\left(x;c\right) & = & \frac{1-c^{2}}{2\pi\left(1+c^{2}-2c\cos x\right)}\\
g_{c}\left(x\right) & = & \frac{1}{\pi}\arctan\left(\frac{1+c}{1-c}\tan\left(\frac{x}{2}\right)\right)\\
r_{c}\left(q\right) & = & 2\arctan\left(\frac{1-c}{1+c}\tan\left(\pi q\right)\right)\\
F\left(x;c\right) & = & \left\{
\begin{array}{ccc}
g_{c}\left(x\right) & & 0\leq x<\pi\\
1-g_{c}\left(2\pi-x\right) & & \pi\leq x\leq2\pi
\end{array}
\right.\\
G\left(q;c\right) & = & \left\{
\begin{array}{ccc}
r_{c}\left(q\right) & & 0\leq q<\frac{1}{2}\\
2\pi-r_{c}\left(1-q\right) & & \frac{1}{2}\leq q\leq1
\end{array}
\right.\end{eqnarray*}
\[h\left[X\right]=\log\left(2\pi\left(1-c^{2}\right)\right).\]
Implementation: scipy.stats.wrapcauchy