# Johnson's SU-distribution

The Johnson's SU-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:

$z=\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)$ Parameters Probability density function Cumulative distribution function $\gamma ,\xi ,\delta >0,\lambda >0$ (real) $-\infty {\text{ to }}+\infty$ ${\frac {\delta }{\lambda {\sqrt {2\pi }}}}{\frac {1}{\sqrt {1+\left({\frac {x-\xi }{\lambda }}\right)^{2}}}}e^{-{\frac {1}{2}}\left(\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)\right)^{2}}$ $\Phi \left(\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)\right)$ $\xi -\lambda \exp {\frac {\delta ^{-2}}{2}}\sinh \left({\frac {\gamma }{\delta }}\right)$ $\xi +\lambda \sinh \left(-{\frac {\gamma }{\delta }}\right)$ ${\frac {\lambda ^{2}}{2}}(\exp(\delta ^{-2})-1)\left(\exp(\delta ^{-2})\cosh \left({\frac {2\gamma }{\delta }}\right)+1\right)$ 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 SU 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.

## Johnson's SB-distribution

N. L. Johnson  firstly proposes the transformation :

$z=\gamma +\delta \log \left({\frac {x-\xi }{\xi +\lambda -x}}\right)$ where $z\sim {\mathcal {N}}(0,1)$ .

Johnson's SB random variables can be generated from U as follows:

$x=\lambda \left(1+\exp \left(-{\frac {\Phi ^{-1}(U)-\gamma }{\delta }}\right)\right)+\xi$ where Φ is the cumulative distribution function of the normal distribution. SB-distribution is convenient to Platykurtic distributions (Kurtosis).

## Applications

Johnson's $S_{U}$ -distribution has been used successfully to model asset returns for portfolio management.

• Hill, I. D.; Hill, R.; Holder, R. L. (1976). "Algorithm AS 99: Fitting Johnson Curves by Moments". Journal of the Royal Statistical Society. Series C (Applied Statistics). 25 (2).
• Jones, M. C.; Pewsey, A. (2009). "Sinh-arcsinh distributions" (PDF). Biometrika. 96 (4): 761. doi:10.1093/biomet/asp053.( Preprint)
• Tuenter, Hans J. H. (November 2001). "An algorithm to determine the parameters of SU-curves in the Johnson system of probability distributions by moment matching". The Journal of Statistical Computation and Simulation. 70 (4): 325–347. doi:10.1080/00949650108812126.