Surf Wiki
Save to docs
general/continuous-distributions

From Surf Wiki (app.surf) — the open knowledge base

Johnson's SU-distribution

Family of probability distributions

Family of probability distributions

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) The Johnson's SU-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 \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 SU 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 ''SB''-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 SB 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 SB-distribution is convenient to Platykurtic distributions (Kurtosis). To simulate SU, 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.

References

References

  1. Johnson, N. L.. (1949). "Systems of Frequency Curves Generated by Methods of Translation". [[Biometrika]].
  2. Johnson, N. L.. (1949). "Bivariate Distributions Based on Simple Translation Systems". [[Biometrika]].
  3. Tsai, Cindy Sin-Yi. (2011). "The Real World is Not Normal". Morningstar Alternative Investments Observer.
  4. LHCb Collaboration. (2022). "Precise determination of the {B}{\mathrm{s}}^{0}–{\overline{B}}{\mathrm{s}}^{0} oscillation frequency". Nature Physics.
Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

continuous-distributions
Want to explore this topic further?

Ask Mako anything about Johnson's SU-distribution — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report