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Johnson SB distribution

Where will you meet this distribution?

Epidemiology
“Fitting human exposure data with the Johnson SB distribution” by MICHAEL R. FLYNN
Forestry
“Plotting Johnson’s SB Distribution using a new parameterization” by Keith Rennolls1 et al

“A new parameterization of Johnson’s SB distribution with application to fitting forest tree diameter data” by Keith Rennolls and Mingliang Wang

“Tree diameter distribution modelling: introducing the logit–logistic distribution” by Mingliang Wang and Keith Rennolls

Shape of Distribution

Basic Properties

Four parameters $\gamma, \delta,\lambda,\xi$ are required.
$\delta>0,\lambda>0$
Continuous distribution defined on bounded range $\xi\leq x \leq \xi+\lambda$
This distribution can be symmetric or asymmetric.

Probability

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

where

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

and $\Phi(\cdot)$ is cumulative distribution function of standard normal distribution.
Probability density function
$f(x)=\frac{\delta}{\lambda\sqrt{2\pi}z(1-z)}\exp\left[-\frac{1}{2}\left(\gamma+\delta\ln\frac{z}{1-z}\right)^2\right]$

How to compute these on Excel.


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A	B
Data	Description
0.5	Value for which you want the distribution
8	Value of parameter Gamma
2	Value of parameter Delta
2	Value of parameter Lambda
2	Value of parameter Xi
=(A2-A5)/A4	Standardized variable z
Formula	Description (Result)
=NORMSDIST(A3+A4*LN(A7/(1-A7)))	Cumulative distribution function for the terms above
=A4EXP(-0.5(A3+A4LN(A7/(1-A7)))^2)/(SQRT(2PI())A5A7*(1-A7))	Probability density function for the terms above

Quantile

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

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 distribution
1.7	Value of parameter Gamma
0.9	Value of parameter Delta
0.9	Value of parameter Lambda
0.9	Value of parameter Xi
Formula	Description (Result)
=A5*EXP((NORMSINV(A2)-A3)/A4)/(1+EXP((NORMSINV(A2)-A3)/A4))+A6	Inverse of the cumulative distribution function for the terms above

Random Numbers

Random number x is generated by inverse function method, which is for uniform random U,
$x=\frac{\lambda\exp\left(\frac{\Phi^{-1}(U)-\gamma}{\delta}\right)}{1+\exp\left(\frac{\Phi^{-1}(U)-\gamma}{\delta}\right)}+\xi$

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

How to generate random numbers on Excel.


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A	B
Data	Description
0.5	Value of parameter Gamma
0.5	Value of parameter Delta
0.5	Value of parameter Lambda
0.5	Value of parameter Xi
Formula	Description (Result)
=A4*EXP((NORMSINV(NTRAND(100))-A2)/A3)/(1+EXP((NORMSINV(NTRAND(100))-A2)/A3))+A5	100 Johnson SB 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.