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# Johnson SU distribution

## Shape of Distribution

### Basic Properties

• Four parameters $\gamma, \delta,\lambda,\xi$ are required (How can you get these).
$\delta>0,\lambda>0$
• Continuous distribution defined on entire range.
• This distribution can be symmetric or asymmetric.

### Probability

• Cumulative distribution function
$F(x)=\Phi\left(\gamma+\delta\sinh^{-1}z\right)$

where

$z=\frac{x-\xi}{\lambda}$

and $\Phi(\cdot)$ is cumulative distribution function of standard normal distribution.

• Probability density function
$f(x)=\frac{\delta}{\lambda\sqrt{2\pi}\sqrt{z^2+1}}\exp\left[-\frac{1}{2}\left(\gamma+\delta\sinh^{-1}z\right)^2\right]$
• How to compute these on Excel.

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A B
Data Description
2.5 Value for which you want the distribution
1 Value of parameter Gamma
4 Value of parameter Delta
3 Value of parameter Lambda
0.9 Value of parameter Xi
Formula Description (Result)
=NTJOHNSONSUDIST(A2,A3,A4,A5,A6,TRUE) Cumulative distribution function for the terms above
=NTJOHNSONSUDIST(A2,A3,A4,A5,A6,FALSE) Probability density function for the terms above

• Function reference : NTJOHNSONSUDIST

### Quantile

• Inverse of cumulative distribution function
$F^{-1}(P)=\lambda\sinh\left(\frac{\Phi^{-1}(P)-\gamma}{\delta}\right)+\xi$

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

• How to compute this on Excel.

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A B
Data Description
0.5 Probability associated with the Johnson SU distribution
1 Value of parameter Gamma
4 Value of parameter Delta
3 Value of parameter Lambda
0.9 Value of parameter Xi
Formula Description (Result)
=NTJOHNSONSUINV(A2,A3,A4,A5,A6) Inverse of the cumulative distribution function for the terms above
• Function reference : NTJOHNSONSUINV

## Characteristics

### Mean – Where is the “center” of the distribution? (Definition)

• Mean of the distribution is given as
$\xi+\text{sign}(\gamma_1)\sigma\frac{\omega-1-m(\omega)}{\omega-1}$

where

$m(\omega)=-2+\sqrt{4+2\left(\omega^2-\frac{\beta_2+3}{\omega^2+2\omega+3}\right)}$
$\omega=\exp\left(\delta^{-2}\right)$

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

• How to compute this on Excel

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A B
Data Description
1 Value of parameter Gamma
4 Value of parameter Delta
3 Value of parameter Lambda
0.9 Value of parameter Xi
Formula Description (Result)
=NTJOHNSONSUMEAN(A2,A3,A4,A5) Mean of the distribution for the terms above
• Function reference : NTJOHNSONSUMEAN

### Standard Deviation – How wide does the distribution spread? (Definition)

• Variance of the distribution is given as
$\lambda(\omega-1)\sqrt{\frac{\omega+1}{2m(\omega)}}$

where

$m(\omega)=-2+\sqrt{4+2\left(\omega^2-\frac{\beta_2+3}{\omega^2+2\omega+3}\right)}$
$\omega=\exp\left(\delta^{-2}\right)$

Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

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Data Description
1 Value of parameter Gamma
4 Value of parameter Delta
3 Value of parameter Lambda
0.9 Value of parameter Xi
Formula Description (Result)
=NTJOHNSONSUSTDEV(A2,A3,A4,A5) Standard deviation of the distribution for the terms above
• Function reference : NTJOHNSONSUSTDEV

### Skewness – Which side is the distribution distorted into? (Definition)

• Skewness of the distribution is given as
$\beta_1=\omega(\omega-1)\frac{[\omega(\omega+2)\sinh 3\Omega+3\sinh\Omega]^2}{2(\omega\cosh 2\Omega+1)^3}$

where

$\omega=\exp\left(\delta^{-2}\right),\;\Omega=\frac{\gamma}{\delta}$
• How to compute this on Excel

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Data Description
1 Value of parameter Gamma
4 Value of parameter Delta
3 Value of parameter Lambda
0.9 Value of parameter Xi
Formula Description (Result)
=NTJOHNSONSUSKEW(A2,A3,A4,A5) Skewness of the distribution for the terms above
• Function reference : NTJOHNSONSUSKEW

### Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

• Kurtosis of the distribution is given as
$\beta_2=\frac{\omega^2(\omega^4+2\omega^3+3\omega^2-3)\cosh 4\Omega+4\omega^2(\omega+2)\cosh 2\Omega+3(2\omega+1)}{2(\omega\cosh 2\Omega+2)^2}-3$

where

$\omega=\exp\left(\delta^{-2}\right),\;\Omega=\frac{\gamma}{\delta}$
• How to compute this on Excel

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Data Description
1 Value of parameter Gamma
4 Value of parameter Delta
3 Value of parameter Lambda
0.9 Value of parameter Xi
Formula Description (Result)
=NTLOGNORMKURT(A2,A3,A4,A5) Kurtosis of the distribution for the terms above
• Function reference : NTJOHNSONSUKURT

## Random Numbers

• Random number x is generated by inverse function method, which is for uniform random U,
$x=\lambda\left(\frac{\Phi^{-1}(U)-\gamma}{\delta}\right)+\xi$
• How to generate random numbers on Excel.

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Data Description
1 Value of parameter Gamma
4 Value of parameter Delta
3 Value of parameter Lambda
0.9 Value of parameter Xi
Formula Description (Result)
=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.