Johnson's SU-distribution

Family of probability distributions

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:^[1]

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

where ${\displaystyle z\sim {\mathcal {N}}(0,1)}$.

Generation of random variables

Let U be a random variable that is uniformly distributed on the unit interval [0, 1]. Johnson's S_U random variables can be generated from U as follows:

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

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

Johnson's SB-distribution

N. L. Johnson^[1] firstly proposes the transformation :

${\displaystyle z=\gamma +\delta \log \left({\frac {x-\xi }{\xi +\lambda -x}}\right)}$

where ${\displaystyle z\sim {\mathcal {N}}(0,1)}$.

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

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

${\displaystyle x=\lambda y+\xi }$

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 ${\displaystyle S_{U}}$-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 ${\displaystyle S_{U}}$-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 ${\displaystyle S_{U}}$-distribution is also used in the modelling of the invariant mass of some heavy mesons in the field of B-physics.^[4]

References

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

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" (PDF). Biometrika. 96 (4): 761. doi:10.1093/biomet/asp053.( 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. MR 1872992. Zbl 1098.62523.