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Johnson SU distribution

Quick view

Johnson SU transforms a normal variable through the inverse hyperbolic sine and produces an unbounded variable. It absorbs skewness and kurtosis that normal cannot represent.

If you are coming from another distribution

SU is the unbounded member of the Johnson system; SB is bounded and SL specializes in positive log-transformable variables. It competes with skew-normal and generalized-normal families.

History and terminology

Johnson proposed SB, SU, and SL in 1949 as a systematic way to approximate distributions by transforming normal variables.

A familiar situation

It is a candidate when values can be positive or negative and tails differ from normal. In calibration, the latent-normal transform can simplify simulation and quantile calculations.

Fitting with care

Four parameters can fit small samples impressively for geometric rather than substantive reasons; validate on held-out data, not only a histogram.

A hyperbolic transformation for tails and skewness

Johnson SU uses the hyperbolic sine to turn a Normal variable into one with no finite bounds. The letters SU denote the unbounded system. Two parameters transform skewness and kurtosis, while two more set location and scale.

The family can produce tails far heavier than the Normal and skewness in either direction. It is a flexible fitting tool rather than a universal physical explanation. Its SB partner covers bounded variables, while Lognormal forms the transition in Johnson’s original system. A practical advantage is checking Normality after applying the estimated transformation. If structure remains on that scale, four-parameter flexibility has not repaired underlying dependence or mixtures.

Decision guide

A good candidate when: data span the real line and require both asymmetry and tail control through a Normal transformation.

Compare it with: Skew-normal, skew-t, or Generalized Normal according to the missing feature. Verify that four-parameter flexibility improves prediction rather than only fit.

References

Johnson SU Distribution: equations and calculator

Distribution defintion

X∼JohnsonSU(ξ,λ,γ,δ)X\sim\mathrm{JohnsonSU}\left(\xi,\lambda,\gamma,\delta\right)

Distribution domain

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Parameters domain and parameters constraints

ξ∈R,λ∈R+,γ∈R,δ∈R+\xi\in\mathbb{R},\lambda\in\mathbb{R}^{+},\gamma\in\mathbb{R},\delta\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=Φ(γ+δsinh⁡−1(z(x)))F_{X}\left(x\right)=\Phi\left(\gamma+\delta\sinh^{-1}(z(x))\right)

Probability density function

fX(x)=δλ2πz(x)2+1exp⁡[−12(γ+δsinh⁡−1(z(x)))2]f_{X}\left(x\right)=\frac{\delta}{\lambda\sqrt{2\pi}\sqrt{z(x)^2+1}}\exp\left[-\frac{1}{2}\left(\gamma+\delta\sinh^{-1}(z(x))\right)^2\right]

Percent point function/Sample

FX−1(u)=λsinh⁡(Φ−1(u)−γδ)+ξF^{-1}_{X}\left(u\right)=\lambda\sinh\left(\frac{\Phi^{-1}(u)-\gamma}{\delta}\right)+\xi

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx\mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′=ξ−λexp⁡δ−22sinh⁡(γδ)\mathrm{Mean}(X)=\mu'_{1}=\xi-\lambda \exp\frac{\delta^{-2}}{2} \sinh\left(\frac{\gamma}{\delta}\right)

Parametric variance

Variance(X)=μ2′−μ1′2=λ22(exp⁡(δ−2)−1)(exp⁡(δ−2)cosh⁡(2γδ)+1)\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{\lambda^2}{2} (\exp(\delta^{-2})-1)\left(\exp(\delta^{-2}) \cosh\left(\frac{2\gamma}{\delta}\right) +1\right)

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=−λ3eδ−2(eδ−2−1)2(eδ−2)(eδ−2+2)sinh⁡(3γδ)+3sinh⁡(2γδ))4Variance(X)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=-\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\mathrm{Variance}(X)^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=λ4(eδ−2−1)2(K1+K2+K3)8Variance(X)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=\frac{\lambda^{4}(e^{\delta^{-2}}-1)^{2}(K_{1}+K_{2}+K_{3})}{8\mathrm{Variance}(X)^{2}}

Parametric median

Median(X)=ξ+λsinh⁡(−γδ)\mathrm{Median}(X)=\xi+\lambda \sinh\left(-\frac{\gamma}{\delta}\right)

Parametric mode

Mode(X)=arg⁡max⁡xfX(x)\mathrm{Mode}(X)=\arg\max_{x}f_{X}\left(x\right)

Additional information and definitions

ξ:Location parameter\xi:\text{Location parameter}
λ:Scale parameter\lambda:\text{Scale parameter}
z(x)=(x−ξ)/λz\left(x\right)=\left(x-\xi\right)/\lambda
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Φ(x):CDF normal standard distribution\Phi\left(x\right):\text{CDF normal standard distribution}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}
K1=(eδ−2)2((eδ−2)4+2(eδ−2)3+3(eδ−2)2−3)cosh⁡(4γδ)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)
K2=4(eδ−2)2((eδ−2)+2)cosh⁡(3γδ)K_{2}=4\left(e^{\delta^{-2}}\right)^{2}\left(\left(e^{\delta^{-2}}\right)+2\right)\cosh\left(\frac{3\gamma}{\delta}\right)
K3=3(2(eδ−2)+1)K_{3}=3\left(2\left(e^{\delta^{-2}}\right)+1\right)