# 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.[1][2] Johnson proposed it as a transformation of the normal distribution:[3]

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

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

## 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:

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

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

## Johnson's SB-distribution

N. L. Johnson [1] firstly proposes the transformation :

${\displaystyle z=\gamma +\delta \log \left({\frac {x-\xi }{\xi +\lambda -x}}\right)}$

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

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

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

where Φ is the cumulative distribution function of the normal distribution. SB-distribution is convenient to Platykurtic distributions (Kurtosis).

## Applications

Johnson's ${\displaystyle S_{U}}$-distribution has been used successfully to model asset returns for portfolio management.[4]

## References

1. Johnson, N. L. (1949). "Systems of Frequency Curves Generated by Methods of Translation". Biometrika. 36 (1/2): 149–176. doi:10.2307/2332539. JSTOR 2332539.
2. Johnson, N. L. (1949). "Bivariate Distributions Based on Simple Translation Systems". Biometrika. 36 (3/4): 297–304. doi:10.1093/biomet/36.3-4.297. JSTOR 2332669.
3. Johnson (1949) "Systems of Frequency Curves...", p. 158
4. Tsai, Cindy Sin-Yi (2011). "The Real World is Not Normal" (PDF). Morningstar Alternative Investments Observer.
• 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.