Johnson's SU-distribution

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Johnson's SU
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)
Variance \frac{\lambda^2}{2} (\exp(\delta^{-2})-1) \left( \exp(\delta^{-2}) \cosh \left(\frac{2\gamma}{\delta} \right) +1 \right)

The Johnson's SU-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:[3]

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.

References

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  2. Lua error in package.lua at line 80: module 'strict' not found.
  3. Johnson (1949) "Systems of Frequency Curves...", p. 158

Additional reading

  • Lua error in package.lua at line 80: module 'strict' not found.
  • Lua error in package.lua at line 80: module 'strict' not found.( Preprint)
  • Lua error in package.lua at line 80: module 'strict' not found.