Johnson's SU-distribution

Johnson's S_U

Probability density function
Cumulative distribution function
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	$\xi -\lambda \exp {\frac {\delta ^{-2}}{2}}\sinh \left({\frac {\gamma }{\delta }}\right)$
Median	$\xi +\lambda \sinh \left(-{\frac {\gamma }{\delta }}\right)$
Variance	${\frac {\lambda ^{2}}{2}}(\exp(\delta ^{-2})-1)\left(\exp(\delta ^{-2})\cosh \left({\frac {2\gamma }{\delta }}\right)+1\right)$
Skewness	$-{\frac {\lambda ^{3}{\sqrt {e^{\delta ^{-2}}}}(e^{\delta ^{-2}}-1)^{2}((e^{\delta ^{-2}})(e^{\delta ^{-2}}+2)\sinh({\frac {3\gamma }{\delta }})+3\sinh({\frac {2\gamma }{\delta }}))}{4\mathrm {Variance} (X)^{1.5}}}$
Excess kurtosis	${\frac {\lambda ^{4}(e^{\delta ^{-2}}-1)^{2}(K_{1}+K_{2}+K_{3})}{8\mathrm {Variance} (X)^{2}}}$ $K_{1}=\left(e^{\delta ^{-2}}\right)^{2}\left(\left(e^{\delta ^{-2}}\right)^{4}+2\left(e^{\delta ^{-2}}\right)^{3}+3\left(e^{\delta ^{-2}}\right)^{2}-3\right)\cosh \left({\frac {4\gamma }{\delta }}\right)$ $K_{2}=4\left(e^{\delta ^{-2}}\right)^{2}\left(\left(e^{\delta ^{-2}}\right)+2\right)\cosh \left({\frac {3\gamma }{\delta }}\right)$ $K_{3}=3\left(2\left(e^{\delta ^{-2}}\right)+1\right)$

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.

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

$y={\left(1+{e}^{-\left(z-\gamma \right)/\delta }\right)}^{-1}$

$x=\lambda y+\xi$

The S_B-distribution is convenient to Platykurtic distributions (Kurtosis). To simulate S_U, sample of code for its density and cumulative density function is available here

Applications

Johnson's $S_{U}$-distribution has been used successfully to model asset returns for portfolio management. Johnson distributions are also sometimes used in option pricing, so as to accommodate an observed volatility smile; see Johnson binomial tree.

An alternative to the Johnson system of distributions is the quantile-parameterized distributions (QPDs). QPDs can provide greater shape flexibility than the Johnson system. Instead of fitting moments, QPDs are typically fit to empirical CDF data with linear least squares.

Further reading

• 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 S_U-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.