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st: Estimating the parameters of the GB2 distribution with incomplete knowledge of sample statistics


From   Julio Estevez <je_123@hotmail.com>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: Estimating the parameters of the GB2 distribution with incomplete knowledge of sample statistics
Date   Sun, 26 Sep 2010 15:32:51 -0500

Hi
I am interested in deducing the parameters of a GB2 distribution given that I know just a couple of  quantiles and sample statistics of the population I am interested in.

I know the mean  of the population of interest. I also know some key quantiles from the left side of the distribution. That is  I know x1, x2, x3  such that F(x1) = 0.1, F(x2) = 0.2, F(x3) = 0.3 been F() the c.d.f. Although I do not know much about the upper tail, from previous research I beleive that the GB2 distribution may be good approximation of the distribution I am looking at.

I beleive I  can use the fact that  GB2 distribution has distribution function (c.d.f.)  F(x) = ibeta(p, q, (x/b)^a/(1+(x/b)^a) ) to estimate the parameters of the distribution function (a,b,p,q)  given my knowledge about the mean and the first qualtiles. Then I can explore the upper tail under the assumption that my data is distributed as a GB2

Can somebody help me to  set up the program to estimate these parameters?  I believe it is "just" a matter of solving a set of 4 nonlinear equations with 4 unknowns, but my programming skills are not that good.

The four equations will look like
mean = b*G(p+1/a)*G(q-1/a)/[G(p)G(q)]F(x1) = 0.1 = ibeta(p, q, (x1/b)^a/(1+(x1/b)^a)F(x2) = 0.2 = ibeta(p, q, (x2/b)^a/(1+(x2/b)^a)F(x3) = 0.3 = ibeta(p, q, (x3/b)^a/(1+(x3/b)^a)
from which the unknowns, a,b, p and q can be retrieved. 
with G(.) been the gamma function

Any help will be greatly appreciated!
Thanks

Julio
 		 	   		  
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