Arcsine Distribution#

Defined over \(x\in\left[0,1\right]\). To get the definition presented in Johnson, Kotz, and Balakrishnan, substitute \(x=\frac{u+1}{2}.\) i.e. \(L=-1\) and \(S=2.\)

\begin{eqnarray*} f\left(x\right) & = & \frac{1}{\pi\sqrt{x\left(1-x\right)}}\\ F\left(x\right) & = & \frac{2}{\pi}\arcsin\left(\sqrt{x}\right)\\ G\left(q\right) & = & \sin^{2}\left(\frac{\pi}{2}q\right)\end{eqnarray*}
\[M\left(t\right)=1 + \sum_{k=1}^\infty \left( \prod_{r=0}^{k-1} \frac{2r + 1}{2r+2} \right) \frac{t^k}{k!}\]
\begin{eqnarray*} \mu_{n}^{\prime} & = & \frac{1}{\pi}\int_{0}^{1} x^{n-1/2}\left(1-x\right)^{-1/2} dx\\ & = & \frac{1}{\pi}B\left(\frac{1}{2},n+\frac{1}{2}\right)=\frac{\left(2n-1\right)!!}{2^{n}n!}\end{eqnarray*}
\begin{eqnarray*} \mu & = & \frac{1}{2}\\ \mu_{2} & = & \frac{1}{8}\\ \gamma_{1} & = & 0\\ \gamma_{2} & = & -\frac{3}{2}\end{eqnarray*}
\[h\left[X\right] = \log(\frac{\pi}{4}) \approx-0.24156447527049044468\]
\[l_{\mathbf{x}}\left(\cdot\right)=N\log\pi+\frac{N}{2}\overline{\log\mathbf{x}}+\frac{N}{2}\overline{\log\left(1-\mathbf{x}\right)}\]

References#

Implementation: scipy.stats.arcsine