Johnson SU Distribution¶

There are two shape parameters \(a\in\mathbb{R}\) and \(b>0\), and the support is \(x\in\mathbb{R}\).

\begin{eqnarray*} f\left(x;a,b\right) & = & \frac{b}{\sqrt{x^{2}+1}}\phi\left(a+b\log\left(x+\sqrt{x^{2}+1}\right)\right)\\ F\left(x;a,b\right) & = & \Phi\left(a+b\log\left(x+\sqrt{x^{2}+1}\right)\right)\\ G\left(q;a,b\right) & = & \sinh\left(\frac{\Phi^{-1}\left(q\right)-a}{b}\right)\end{eqnarray*}