Johnson's SU-distribution

Template:Short description Template:Expert needed Template:Probability 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:

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

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

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