Johnson's SU-distribution

Short description: Family of probability distributions

Johnson's S_U

Probability density function
Cumulative distribution function
Parameters	$\gamma ,\xi ,\delta >0,\lambda >0$ (real)
Support	$-\mathrm{\infty }\text{ to }+\mathrm{\infty }$
PDF	$\frac{\delta }{\lambda \sqrt{2\pi }}\frac{1}{\sqrt{1+{\left(\frac{x-\xi }{\lambda }\right)}^2}}{e}^{-\frac{1}{2}{(\gamma +\delta {\sinh }^{-1}\left(\frac{x-\xi }{\lambda }\right))}^2}$
CDF	$\mathrm{\Phi }(\gamma +\delta {\sinh }^{-1}\left(\frac{x-\xi }{\lambda }\right))$
Mean	$\xi -\lambda \exp \frac{{\delta }^{-2}}{2}\sinh \left(\frac{\gamma }{\delta }\right)$
Median	$\xi +\lambda \sinh (-\frac{\gamma }{\delta })$
Variance	$\frac{{\lambda }^2}{2}(\exp ({\delta }^{-2})-1)(\exp ({\delta }^{-2})\cosh \left(\frac{2\gamma }{\delta }\right)+1)$
Skewness	$-\frac{{\lambda }^3\sqrt{e^{{\delta }^{-2}}}(e^{{\delta }^{-2}}-1{)}^2((e^{{\delta }^{-2}})(e^{{\delta }^{-2}}+2)\sinh (\frac{3\gamma }{\delta })+3\sinh (\frac{2\gamma }{\delta }))}{4(\mathrm{Variance}X{)}^{1.5}}$
Kurtosis	$\frac{{\lambda }^4(e^{{\delta }^{-2}}-1{)}^2(K_1+K_2+K_3)}{8(\mathrm{Variance}X{)}^2}$ $K_1={\left(e^{{\delta }^{-2}}\right)}^2({\left(e^{{\delta }^{-2}}\right)}^4+2{\left(e^{{\delta }^{-2}}\right)}^3+3{\left(e^{{\delta }^{-2}}\right)}^2-3)\cosh \left(\frac{4\gamma }{\delta }\right)$ $K_2=4{\left(e^{{\delta }^{-2}}\right)}^2(\left(e^{{\delta }^{-2}}\right)+2)\cosh \left(\frac{3\gamma }{\delta }\right)$ $K_3=3(2\left(e^{{\delta }^{-2}}\right)+1)$

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 $z\sim {𝒩}(0,1)$.

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:

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

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 $z\sim {𝒩}(0,1)$.

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

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

$x=\lambda y+\xi$

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

Applications

Johnson's $S_U$-distribution has been used successfully to model asset returns for portfolio management. This comes as a superior alternative to using the Normal distribution to model asset returns. An R package, JSUparameters, was developed in 2021 to aid in the estimation of the parameters of the best-fitting Johnson's $S_U$-distribution for a given dataset. 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.

Johnson's $S_U$-distribution is also used in the modelling of the invariant mass of some heavy mesons in the field of B-physics.

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". Biometrika 96 (4): 761. doi:10.1093/biomet/asp053. http://oro.open.ac.uk/22510/1/sinhasinh.pdf. ( 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.