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:


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

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:
 * $$y={\left(1+{e}^{-\left(z-\gamma\right) /\delta }\right)}^{-1}$$
 * $$ x=\lambda y +\xi $$

The SB-distribution is convenient to Platykurtic distributions (Kurtosis). To simulate SU, 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.