Johnson's SU-distribution

{{DISPLAYTITLE:Johnson's S_U-distribution}}{{Expert needed |statistics |2= |talk= |reason=completion to reasonable standard for probability distributions|date=November 2012 }}

{{Probability distribution

| name = Johnson's S_U

| type = continuous

| pdf_image = File:JohnsonSU.png

| cdf_image = File:JohnsonSU CDF.png

| notation =

| parameters = $\gamma, \xi, \delta > 0, \lambda > 0$ (real)

| support = $-\infty \text{ to } +\infty$

| pdf = $\frac{\delta}{\lambda\sqrt{2\pi}} \frac{1}{\sqrt{1 + \left(\frac{x-\xi}{\lambda}\right)^2}} e^{-\frac{1}{2}\left(\gamma+\delta \sinh^{-1} \left(\frac{x-\xi}{\lambda}\right)\right)^2}$

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

| mean = $\xi - \lambda \exp \frac{\delta^{-2}}{2} \sinh\left(\frac{\gamma}{\delta}\right)$

| median = $\xi + \lambda \sinh \left( - \frac{\gamma}{\delta} \right)$

| mode =

| variance = $\frac{\lambda^2}{2} (\exp(\delta^{-2})-1) \left( \exp(\delta^{-2}) \cosh \left(\frac{2\gamma}{\delta} \right) +1 \right)$

| skewness = $-\frac{\lambda^3\sqrt{e^{\delta^{-2}}}(e^{\delta^{-2}}-1)^{2}((e^{\delta^{-2}})(e^{\delta^{-2}}+2)\sinh(\frac{3\gamma}{\delta})+3\sinh(\frac{2\gamma}{\delta}))}{4(\operatorname{Variance}X)^{1.5}}$

| kurtosis = $\frac{\lambda^4(e^{\delta^{-2}}-1)^2(K_{1}+K_2+K_3)}{8(\operatorname{Variance}X)^2}$
$K_{1}=\left( e^{\delta^{-2}} \right)^{2}\left( \left( e^{\delta^{-2}} \right)^{4}+2\left( e^{\delta^{-2}} \right)^{3}+3\left( e^{\delta^{-2}} \right)^{2}-3 \right)\cosh\left( \frac{4\gamma}{\delta} \right)$
$K_2=4\left( e^{\delta^{-2}} \right)^2 \left( \left( e^{\delta^{-2}} \right)+2 \right)\cosh\left( \frac{3\gamma}{\delta} \right)$
$K_3=3\left( 2\left( e^{\delta^{-2}} \right)+1 \right)$

| entropy =

| pgf =

| mgf =

| char =

}}

The Johnson's S_U-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949.{{cite journal |author-link=Norman Lloyd Johnson |last=Johnson |first=N. L. |date=1949 |title=Systems of Frequency Curves Generated by Methods of Translation|journal=Biometrika |volume=36 |issue=1/2 |pages=149–176 |jstor=2332539 |doi=10.2307/2332539}}{{cite journal |author-link=Norman Lloyd Johnson |last=Johnson |first=N. L. |date=1949 |title=Bivariate Distributions Based on Simple Translation Systems |journal=Biometrika |volume=36 |issue=3/4 |pages=297–304 |jstor=2332669 |doi=10.1093/biomet/36.3-4.297}} 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 S_U 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 ''S<sub>B</sub>''-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 S_B 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 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 [https://blogs.sas.com/content/iml/2020/01/20/johnson-sb-distribution.html here]

Applications

Johnson's $S_{U}$ -distribution has been used successfully to model asset returns for portfolio management.{{Cite journal|last=Tsai|first=Cindy Sin-Yi|date=2011|title=The Real World is Not Normal|url=http://morningstardirect.morningstar.com/clientcomm/iss/Tsai_Real_World_Not_Normal.pdf|journal=Morningstar Alternative Investments Observer}}

This comes as a superior alternative to using the Normal distribution to model asset returns. An R package, [https://CRAN.R-project.org/package=JSUparameters 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.As an example, see: {{cite journal | doi = 10.1038/s41567-021-01394-x | author = LHCb Collaboration | title = Precise determination of the ${B}_{\mathrm{s}}^{0}$ – ${\overline{B}}_{\mathrm{s}}^{0}$ oscillation frequency | journal = Nature Physics | volume = 18 | year = 2022 | pages = 1-5| doi-access = free | arxiv = 2104.04421 }}

Johnson's SU-distribution

Generation of random variables

Johnson's ''S<sub>B</sub>''-distribution

Applications

References

Further reading