Johnson's SU-distribution

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:

: $z=\gamma+\delta \sinh^ \left(\frac\right)$

where $z \sim \mathcal(0,1)$.

Generation of random variables

Let ''U'' be a random variable that is uniformly distributed on the unit interval  , 1 Johnson's ''S_U'' random variables can be generated from ''U'' as follows:

: $x = \lambda \sinh\left( \frac \right) + \xi$

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

Johnson's ''S_B''-distribution

N. L. Johnson firstly proposes the transformation :

: $z=\gamma+\delta \log \left(\frac\right)$

where $z \sim \mathcal(0,1)$.

Johnson's ''S_B'' random variables can be generated from ''U'' as follows:
: $y=^$
: $x=\lambda y +\xi$

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 availabl
here

Applications

Johnson's $S_$-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 $S_$-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 $S_$-distribution is also used in the modelling of the invariant mass of some heavy mesons in the field of B-physics.As an example, see:

References

Further reading

*
*
Preprint

*

{{ProbDistributions, continuous-infinite

Continuous distributions