包裹柯西分布#
有一个形状参数 \(c\in\left(0,1\right)\)，其支持范围为 \(x\in\left[0,2\pi\right]\).
\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*}
\[h\left[X\right]=\log\left(2\pi\left(1-c^{2}\right)\right).\]