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Johnson SU Distribution

Where do you meet this distribution?​

Shape of Distribution​

Basic Properties​

  • Four parameters γ,δ,λ,ξ\gamma, \delta,\lambda,\xi are required (How can you get these).

    δ>0,λ>0\delta>0,\lambda>0

  • Continuous distribution defined on entire range.

  • This distribution can be symmetric or asymmetric.

Probability​

  • Cumulative distribution function

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

    where

    z=x−ξλz=\frac{x-\xi}{\lambda}

    and Φ(⋅)\Phi(\cdot)") is cumulative distribution function of standard normal distribution.

  • Probability density function

    f(x)=δλ2πz2+1exp⁡[−12(γ+δsinh⁡−1z)2]f(x)=\frac{\delta}{\lambda\sqrt{2\pi}\sqrt{z^2+1}}\exp\left[-\frac{1}{2}\left(\gamma+\delta\sinh^{-1}z\right)^2\right]

  • How to compute these on Excel.

AB
1DataDescription
22.5Value for which you want the distribution
31Value of parameter Gamma
44Value of parameter Delta
53Value of parameter Lambda
60.9Value of parameter Xi
7FormulaDescription (Result)
8=NTJOHNSONSUDIST(A2,A3,A4,A5,A6,TRUE)Cumulative distribution function for the terms above
9=NTJOHNSONSUDIST(A2,A3,A4,A5,A6,FALSE)Probability density function for the terms above

Sample distribution

Quantile​

  • Inverse of cumulative distribution function

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

    where Φ(⋅)\Phi(\cdot) is cumulative distribution function of standard normal distribution.

  • How to compute this on Excel.

AB
1DataDescription
20.5Probability associated with the Johnson SU distribution
31Value of parameter Gamma
44Value of parameter Delta
53Value of parameter Lambda
60.9Value of parameter Xi
7FormulaDescription (Result)
8=NTJOHNSONSUINV(A2,A3,A4,A5,A6)Inverse of the cumulative distribution function for the terms above

Characteristics​

Mean -- Where is the "center" of the distribution? (Definition)​

  • Mean of the distribution is given as

    ξ+sign(γ1)σω−1−m(ω)ω−1\xi+\text{sign}(\gamma_1)\sigma\frac{\omega-1-m(\omega)}{\omega-1}

    where

    m(ω)=−2+4+2(ω2−β2+3ω2+2ω+3)m(\omega)=-2+\sqrt{4+2\left(\omega^2-\frac{\beta_2+3}{\omega^2+2\omega+3}\right)} ω=exp⁡(δ−2)\omega=\exp\left(\delta^{-2}\right)

    and γ1\gamma_1 is skewness of the distribution (see below)

  • How to compute this on Excel

AB
1DataDescription
21Value of parameter Gamma
34Value of parameter Delta
43Value of parameter Lambda
50.9Value of parameter Xi
6FormulaDescription (Result)
7=NTJOHNSONSUMEAN(A2,A3,A4,A5)Mean of the distribution for the terms above

Standard Deviation -- How wide does the distribution spread? (Definition)​

  • Variance of the distribution is given as

    λ(ω−1)ω+12m(ω)\lambda(\omega-1)\sqrt{\frac{\omega+1}{2m(\omega)}}

    where

    m(ω)=−2+4+2(ω2−β2+3ω2+2ω+3)m(\omega)=-2+\sqrt{4+2\left(\omega^2-\frac{\beta_2+3}{\omega^2+2\omega+3}\right)} ω=exp⁡(δ−2)\omega=\exp\left(\delta^{-2}\right)

    Standard Deviation is a positive square root of Variance.

  • How to compute this on Excel

AB
1DataDescription
21Value of parameter Gamma
34Value of parameter Delta
43Value of parameter Lambda
50.9Value of parameter Xi
6FormulaDescription (Result)
7=NTJOHNSONSUSTDEV(A2,A3,A4,A5)Standard deviation of the distribution for the terms above

Skewness -- Which side is the distribution distorted into? (Definition)​

  • Skewness of the distribution is given as

    β1=ω(ω−1)[ω(ω+2)sinh⁡3Ω+3sinh⁡Ω]22(ωcosh⁡2Ω+1)3\beta_1=\omega(\omega-1)\frac{[\omega(\omega+2)\sinh 3\Omega+3\sinh\Omega]^2}{2(\omega\cosh 2\Omega+1)^3}

    where

    ω=exp⁡(δ−2),Ω=γδ\omega=\exp\left(\delta^{-2}\right),\Omega=\frac{\gamma}{\delta}

  • How to compute this on Excel

AB
1DataDescription
21Value of parameter Gamma
34Value of parameter Delta
43Value of parameter Lambda
50.9Value of parameter Xi
6FormulaDescription (Result)
7=NTJOHNSONSUSKEW(A2,A3,A4,A5)Skewness of the distribution for the terms above

Kurtosis -- Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)​

  • Kurtosis of the distribution is given as

    β2=ω2(ω4+2ω3+3ω2−3)cosh⁡4Ω+4ω2(ω+2)cosh⁡2Ω+3(2ω+1)2(ωcosh⁡2Ω+2)2−3\beta_2=\frac{\omega^2(\omega^4+2\omega^3+3\omega^2-3)\cosh 4\Omega+4\omega^2(\omega+2)\cosh 2\Omega+3(2\omega+1)}{2(\omega\cosh 2\Omega+2)^2}-3

    where

    ω=exp⁡(δ−2),Ω=γδ\omega=\exp\left(\delta^{-2}\right),\Omega=\frac{\gamma}{\delta}

  • How to compute this on Excel

AB
1DataDescription
21Value of parameter Gamma
34Value of parameter Delta
43Value of parameter Lambda
50.9Value of parameter Xi
6FormulaDescription (Result)
7=NTJOHNSONSUKURT(A2,A3,A4,A5)Kurtosis of the distribution for the terms above

Random Numbers​

  • Random number x is generated by inverse function method, which is for uniform random U,

    x=λ(Φ−1(U)−γδ)+ξx=\lambda\left(\frac{\Phi^{-1}(U)-\gamma}{\delta}\right)+\xi

  • How to generate random numbers on Excel.

AB
1DataDescription
2100Number of random numbers to generate
31Value of parameter Gamma
44Value of parameter Delta
53Value of parameter Lambda
60.9Value of parameter Xi
7FormulaDescription (Result)
8=NTRANDJOHNSONSU(100,A2,A3,A4,A5,0)100 Johnson SU deviates based on Mersenne-Twister algorithm for which the parameters above

Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A7:A106 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions​

Reference​