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st: Johnson SB distribution


From   "Demetris Christodoulou" <D.Christodoulou@econ.usyd.edu.au>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: Johnson SB distribution
Date   Thu, 20 Nov 2008 06:45:56 +1100

First of all, please accept my apologies for the previous null posting.

My question revisits a previous posting dated 05 Dec 2003 when Nick Cox and Stephen Jenkins first advised me on a question related to the Johnson SB distribution (http://www.stata.com/statalist/archive/2003-12/msg00208.html). I was quite sure that there were more postings on this but I could not find them.

A standardised Johnson SB is:
>	Y = (X - xi) / lambda 
and a unit normal Z can be represented as:
>	Z = gamma + delta * logit Y 
Therefore: 
>	Y = invlogit((Z - gamma)/delta) 
>	X = xi + lambda * invlogit((Z - gamma)/delta) 

In Stata terms, I could generate a Johnson SB by:
. set obs 100000
. gen X_SB = xi + lambda * invlogit((invnorm(uniform()) - gamma)/delta)

For instance, generate the X(xi, lambda, gamma, delta) = X(-1, 1, 3, 5):
. gen X_SB = -1 + 2 * invlogit((invnorm(uniform()) - 3)/5)

I expected that the variable X_SB would be distributed from -1 to 1, whereas the result is much narrower range.
Is my logic wrong or is it something else?

many thanks, Demetris

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