Johnson's SU-distribution

Johnson's S_U

Parameters	$\gamma, \xi, \delta > 0, \lambda > 0$ (real)
Support	$-\infty \text{ to } +\infty$
PDF	$\frac{\delta}{\lambda\sqrt{2\pi}} \frac{1}{\sqrt{1 + \left(\frac{x-\xi}{\lambda}\right)^2}} e^{-\frac{1}{2}\left(\gamma+\delta \sinh^{-1} \left(\frac{x-\xi}{\lambda}\right)\right)^2}$
CDF	$\Phi \left(\gamma + \delta \sinh^{-1} \left( \frac{x-\xi}{\lambda} \right) \right)$
Mean	$\xi - \lambda \exp \frac{\delta^{-2}}{2} \sinh\left(\frac{\gamma}{\delta}\right)$
Variance	$\frac{\lambda^2}{2} (\exp(\delta^{-2})-1) \left( \exp(\delta^{-2}) \cosh \left(\frac{2\gamma}{\delta} \right) +1 \right)$

The Johnson's S_U-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949. Johnson proposed it as a transformation of the normal distribution:

$z=\gamma+\delta \sinh^{-1} \left(\frac{x-\xi}{\lambda}\right)$

where $z \sim \mathcal{N}(0,1)$.

Generation of random variables

Let U be a random variable that is uniformly distributed on the unit interval [0, 1]. Johnson's S_U random variables can be generated from U as follows:

$x = \lambda \sinh\left( \frac{ \Phi^{ -1 }( U ) - \gamma }{ \delta } \right) + \xi$

where Φ is the cumulative distribution function of the normal distribution.

Additional reading

• ( Preprint)