Johnson's SU-distribution

Johnson's *S_U*
Parameters	$\gamma, \xi, \delta > 0, \lambda > 0$ (real)
Support	$-\infty \text{ to } +\infty$
PDF	$\frac{\delta}{\lambda\sqrt{2\pi}} \frac{1}{\sqrt{1 + \left(\frac{x-\xi}{\lambda}\right)^2}} e^{-\frac{1}{2}\left(\gamma+\delta \sinh^{-1} \left(\frac{x-\xi}{\lambda}\right)\right)^2}$
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 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:^[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 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.

References

↑ Johnson, N. L. (1949). "Systems of Frequency Curves Generated by Methods of Translation". Biometrika 36 (1/2): 149–176. JSTOR 2332539.
↑ Johnson, N. L. (1949). "Bivariate Distributions Based on Simple Translation Systems". Biometrika 36 (3/4): 297–304. JSTOR 2332669.
↑ Johnson (1949) "Systems of Frequency Curves...", p. 158

Additional 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". 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.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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