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Johnson SU distribution

Where do you meet this distribution?

Finance, Economics : Value at Risk(VaR) model
“Estimation of Value at Risk Using Johnson Su-normal 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

Sample distribution

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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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)
=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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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)
=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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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)
=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.

NtRand Functions

If you already have parameters of the distribution
- Generating random numbers based on Mersenne Twister algorithm: NTRANDJOHNSONSU
- Computing probability : NTJOHNSONSUDIST
- Computing quantile : NTJOHNSONSUINV
- Computing mean : NTJOHNSONSUMEAN
- Computing standard deviation : NTJOHNSONSUSTDEV
- Computing skewness : NTJOHNSONSUSKEW
- Computing kurtosis : NTJOHNSONSUKURT
- Computing moments above at once : NTJOHNSONSUMOM
If you know mean and standard deviation of the distribution
- Estimating parameters of the distribution:NTJOHNSONSUPARAM