Johnson SU distribution

Johnson $S_U$

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

The Johnson $S_U$ 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:

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

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 $S_U$ random variables can be generated from U as follows:

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

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

Additional reading

• I. D. Hill, R. Hill and R. L. Holder, Algorithm AS 99: Fitting Johnson Curves by Moments, Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 25, No. 2 (1976)
• Jones, M. C.; Pewsey, A. (2009). "Sinh-arcsinh distributions". Biometrika 96 (4): 761. doi:10.1093/biomet/asp053.  ( Preprint)