Johnson's SU-distribution explained

The Johnson's S_U-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949.^{[1] [2]} Johnson proposed it as a transformation of the normal distribution:

z = \gamma + \delta \sinh ^{-1} \left( (x - \xi)/(λ) \right)

where

z \sim l{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 = λ \sinh\left( (\Phi ( 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( (x - \xi)/(\xi +λ- x) \right)

where

z \sim l{N}(0,1) .

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

y = {\left( 1+ {e} ^{-\left(z-\gamma\right)} \right)} ^{-1} x =λ 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.^[3] 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.^[4]

Further reading

• I. D. . Hill . R. . Hill . R. L. . Holder . Algorithm AS 99: Fitting Johnson Curves by Moments . Journal of the Royal Statistical Society. Series C (Applied Statistics) . 25 . 2 . 1976.
• Jones . M. C. . Pewsey . A. . 10.1093/biomet/asp053 . Sinh-arcsinh distributions . Biometrika . 96 . 4 . 761 . 2009 . (Preprint)
• Tuenter . Hans J. H. . 10.1080/00949650108812126 . 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 . November 2001. 1872992 . 1098.62523.

Notes and References

1. Norman Lloyd Johnson . Johnson . N. L. . 1949 . Systems of Frequency Curves Generated by Methods of Translation. . 36 . 1/2 . 149–176 . 2332539 . 10.2307/2332539.
2. Norman Lloyd Johnson . Johnson . N. L. . 1949 . Bivariate Distributions Based on Simple Translation Systems . . 36 . 3/4 . 297–304 . 2332669 . 10.1093/biomet/36.3-4.297.
3. Tsai. Cindy Sin-Yi. 2011. The Real World is Not Normal. Morningstar Alternative Investments Observer.
4. As an example, see: 10.1038/s41567-021-01394-x . LHCb Collaboration . Precise determination of the {B} ^{0} _{s} –{\overline{B}} ^{0} _{s} oscillation frequency . Nature Physics . 18 . 2022 . 1-5. free . 2104.04421 .