Johnson SU distribution

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Johnson S_U
Parameters  \gamma, \xi, \delta > 0, \lambda > 0 (real)
Support  -\infty  \text{ to } +\infty
PDF \frac{\delta}{\lambda\sqrt{2\pi}}\frac{1}{\sqrt{1 + (\frac{x-\xi}{\lambda})^{2}}}e^{-\frac{1}{2}(\gamma+\delta \sinh^{-1}(\frac{x-\xi}{\lambda}))^{2}}
CDF \Phi (\gamma + \delta \sinh^{-1}(\frac{x-\xi}{\lambda}))
Mean \xi - \lambda \exp{\frac{\delta^{-2}}{2}} \sinh(\frac{\gamma}{\delta})
Variance \frac{\lambda^2}{2}(\exp{(\delta^{-2})}-1)(\exp{(\delta^{-2})}\cosh{(\frac{2\gamma}{\delta})}+1)

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

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

Where z \sim \mathcal{N}(0,1)

Generation of random variables[edit]

Let U be a random variable that is uniformly distributed on the unit interval [0, 1]. Johnson 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.


  1. ^ Johnson, N. L. (1949) Systems of frequency curves generated by methods of translation. Biometrika 36: 149–176 JSTOR 2332539
  2. ^ N.L. Johnson, Table to facilitate fitting SU frequency curves, Biometrika 1949

Additional reading[edit]

  • I. D. Hill, R. Hill and R. L. Holder, Algorithm AS 99: Fitting Johnson Curves by Moments, Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 25, No. 2 (1976)
  • Jones, M. C.; Pewsey, A. (2009). "Sinh-arcsinh distributions". Biometrika 96 (4): 761. doi:10.1093/biomet/asp053.  edit( Preprint)